does anyone have some thoughts on this:
Let G be a group and K a normal subgroup in G of finite index. Then the set X of all automorphisms phi of G that fix K (meaning phi(K)=K, so not necessarily pointwise) has finite index in Aut G.
Is this even true? I am trying to work out a larger problem and this would be the final step but I am not sure this is even correct.
I tried explaining it by using that the inner automorphisms of G (Int G) are a subset of X (since K is normal). Therefore, if [Aut G:Int G] is finite, so must be [Aut G:X]. But again, I'm not sure this holds...
Thanks very much for your help. I really appreciate it!