The order of H is 2, so it has two elements. Because H is a normal subgroup, it is a subgroup in particular, hence it must contain the identity element.

Hence, H = {e,a} where a is some element of order 2 (there can only be one element of order 1, and the order of the non-trivial element must divide 2, hence it must be 2).

Now since H is a normal subgroup, by definition, for all g in G. In particular, is either e or a.

Note that is an isomorphism (it is an automorphism, actually). This means that it must preserve the order of the element.

Hence, , or else we would have a contradiction.

Thus, for all g in G.

Hence, a commutes with every element in G. Since e does this too, H is a subset of the center of G.

Clearly, since a has order 2, this is a subgroup of the center of G.