Need help to find a reputable company that can accommodate a private, custom tour or at minimum provide transportation for a group of 8 to visit both Avila and Segovia on the same day. Would need transportation starting in Madrid and returning to Madrid at the end of the day. I realize this will be a long day, but trains and busses aren't an option for our group. I have searched and I have only found tour companies that are very structured and can't do both towns in one day. We want to remain flexible with the time we are in each town in case there are especially interesting sites. Travel dates are in late November.
Check out Spanish Trails. They offer a small group tour that goes to Segovia and Pedraza - we took that tour, but they will send you a proposal for a private tour. They arranged a private tour for us to Tarragona (out of Barcelona), which was outstanding.
I recommend starting in Avila, which I am not alone in considering the much less interesting of the two cities. It would be a shame to feel you need to leave Segovia to allow some fixed amount of time in Avila, which really doesn't have a great deal to see.
If your concern about public transportation is due to mobility issues within your group, be aware that the major thing to do in Avila is walk on the wall, which involves climbing and descending steps.