I'm trying to do this methodically. Let

be a non-abelian group of order 42. The Sylow Theorem says that

has a normal subgroup of order 7, and the number of Sylow-3 subgroups is either 1 or 7.

If

, then

, so

and is normal because its index is 2. If we look for homomorphisms

, we find the trivial one (which gives

), or if

and

, then

,

, and

. This gives us three non-abelian groups of order 42, namely

for

. What is the best way to determine whether these are all unique up to isomorphism and, furthermore, whether they are isomorphic to more familiar groups?

Next, if

, then we look for homomorphisms

. If

and

, then

is a valid homomorphism, so

, which I will call

because I think these are called Frobenius groups. So now I want to look for homomorphisms

. Aside from the trivial one, which gives

, I don't know whether there are any other ones because I don't understand the structure of

, so in particular I don't know whether its automorphism group has any elements of order 2.

Thanks in advance!