It could be a servo malfunction or perhaps a sensor controlling the servo that engages the next clutch for the second gear. But for something like this to fail it has to be really a rare case, as auto transmissions are mean to take all the brunt of engine and drive-train workload and for a transmission to fail, is also a rare case. Auto transmission are run by a torque converter and are overall lubricated by the automatic transmission fluid. For a transmission to fail, it is either neglected or trashed, e.g. never had a service, never had a fluid change, in other words - checked the fluid level. And then there is the trashing, e.g. trashed without service, low in fluid and hard manual change (revving the engine and then slam it into 'Drive' to do wheelies). The first thing you need to do it check the fluid level, what colour is it, dark wine colour is very old (and dirty), if it is light colour fluid (like the same colour as the new fluid in the container), then the fluid is okay, we move to the next step. Many transmissions have filters inside the housing and these trap small particles of clutch and metal fragments. If you're a mechanic you can start here. if you're not, don't bother, as transmission work is not for you. Let's assume you know enough about mechanical work and have a set of sockets, a wrench and a torque wrench (a specialized tool to tighten bolts and nuts to the correct tension). So, work to transmissions are to be done when cold, that is when the car hasn't been running on the road, so the fluid is not hot, Some transmissions have a draining plug. If so, drain the fluid in a white 6 litre container (white so that you can see the colour of the fluid and whether there are fragments of clutch material at the bottom of the container, later. Do not do anything else unless you can see these symptoms: fluid has burned smell, fluid has a dark wine colour, and when feeling the fluid between the fingers it feels sticky rather than smooth. Now, use another white container and pour the fluid into it, noting the colour of the fluid leaving the first container. If it is still darker and you can hardly see the lip of the first container through the fluid being poured into the second container then that fluid had never been changed or it is a result of being fouled by hard flooring and changing gears while the engine is at high revs. Either way this is the result of the burned smell. If you noticed any of these, it's time to remove the transmission pan,
Remove all bolts, then carefully remove the pan flat to avoid spilling the remnant fluid on you and also to avoid disturbing the sediments at the bottom of the pan. Once removed, pour whatever fluid is left in the pan in a clean white container and inspect the pan for sediments, broken clutch material and small metallic grains or fragments and the smell. This will indicate that the transmission will have to have a major overhaul but it's not necessary because this also indicates that to neglect a transmission would mean to destroy it and eventually it will be destroyed by the way you drive. What you need to do now is to see if there are any signs of a filter attached under the transmission. remove it and clean with mineral turps, solvent or kerosene. Allow to drain and when dry dip it into a transmission fluid before replacing it back on the transmission. Add a new pan (aka sump) gasket and replace the bolts, tighten by hand, then using a suitable socket and a hand wrench tighten them just so they seat hard on, then use the torque wrench, set it to 20 ft lb and start tightening each bolt in a criss-cross fashion - one bolt here then the other bolt across the pan, then go back to the second bolt where you tightened the first and tighten that one, then go back to the second bolt and tighten the one next to the second bolt and so on, eventually reaching the fist bolt. Be sure you have a lot of concentration of which bolt you have to tighten next and always remember to re-set the pin on the torque wrench to make it click when the set tension has reached. Over tightening the pan or sump will damage it. So, stay focused on the tightening sequence. if unsure and you might lose track, then number them with a texta or chalk. Then set the torque wrench to (I think 30 ft lb) and tighten each bolt, starting from the first bolt and repeat the same sequence until all the bolts are tightened. Note that I am only guessing the torque of 30 ft lb for the type of bolts I think your transmission have, but the bolts could be bigger or smaller, so, to be sure, you should refer this subject to a mechanic, or search online for bolt tightening charts. Refill the transmission then run the engine and check the fluid level. Top it up to the required level if necessary. Then do several slow forward - reverse changes to allow the fluid to disperse into critical components, valves, servos, etc., then again check the fluid level and road test the car. Hopefully it should change gears. if not, then it may be too late to save your day and a major overhaul is necessary. If not, a change-over transmission would be cheaper. See how you will do with these home repairs first. Also, ask about what criss-crossing bolt tightening sequence means before you engage in repairs. Cheers.
Photo is a standard torque wrench for most mechanical work.