My questions are:

Is there any group, which cannot be written as the quotient of a residually finite group by an amenable normal subgroup? Is it possible for large classes of groups?

and

Is there a group, such that every extension by an amenable group is split?

quotientof a residually finite group by an amenable subgroup (but that takes some time to explain). I was wondering what kind of condition one gets in the more rigid setup of ordinary group theory. Burnside groups are not known to be sofic, so Mark's comment is very interesting. – Andreas Thom Oct 4 '10 at 16:32