Of the three responses so far, only David R seems to have read carefully enough to have understood the question.

Softwareguy’s recommendation to “Never dial a connection” would “break” my automatic mail fetch, where all my wife has to do is click on the “Both check and send mail” icon, and the rest is automatic. This is a non-solution.

Technical tries to tell me that " avast! does not have a ‘dialer’… It can’t dial in any way." When I know very well that Avast can trigger the windows dialler, just the same way that any other program needing access to the internet can dial. No, it does not have its own dialer, but it certainly can force Windows’ built-in dialer to dial out.

I will try David R’s suggestion to lengthen the timout on the hangup, and see if that settles things down.