Fully characteristic subgroups

[math8]

Let G=Z_2XS_3 (Z_2:cyclic group of order 2; S_3: Symmetric group on 3) . Show Center of G, Z(G) is not a fully characteristic (or invariant) subgroup of G.

Apparently, Z(G)=Z_2
I know that I need to show that there exists an endomorphism g from G to G such that g(Z_2) is not contained in Z_2.
But I am not sure how.

Also, to prove that every fully characteristic subgroup H is also characteristic, I now how to show that for every automorphism p in Aut(G), p(H) is contained in H, but for some reason I don't see why H is contained in p(H).

 

[Dick]

There are subgroups of S_3 that are isomorphic to Z_2. Map the center to one of those. For the second question, p^(-1) is also in Aut(G).