My understanding is that ASP.NET sites run within an AppPool process on IIS. This AppPool is specifically conigured for a number of things, one of which is the .NET version. It's possible to have multiple AppPools on IIS, giving you the ability to run .NET2 sites alongside .NET3.5 and .NET4 sites. However, each site must be configured with the correct AppPool that you want to use.
It's not clear to me what you mean by running an ASP.NET application 'inside' another one. I would suspect that if you have a site that sits under another site, it would reside in the same AppPool as the parent, and therefore would share the .NET version, security credentials etc.