Message: Unfortunately Google Play Store has stopped The problem is happening on a very old HTC Sensation running Android 4.0.3, and all the help found online refers to more modern versions.

In Apps -> Google Play Store, I forced stop, cleared data and cache, rebooted but it didn't cure the problem. I also performed a factory reset without success.
Someone in the Google Play Help Forum advised me to call the Play support team which I did.  They sent me an email, I followed the procedure but it didn't fix anything. They told me to contact HTC if the bug still persisted, which I did a while ago but no reply, so far.
I have solved this problem and I am posting the solution here in case someone else would have the same problem.
I downloaded, installed and ran Lucky Patcher apk file V6.4, for Android 4.0.3, 4.0.4 from here and it did the trick.