Example of a subgroup of index two which contains a non square element

Question

If a finite group contains a subgroup H of index two, then every element of the group which is a square belongs to H. Is there a (simple) counterexample showing that not all the elements of H are necessarily squares?

Answer 1

Even easier... In the Klein 4-group, there are three subgroups of index 2, but only one element is a square. (It's the identity.)

Answer 2

Try $G=\mathbb{Z}_4\oplus \mathbb{Z}_4$ and $H=\mathbb{Z}_4\oplus \{0,2\}$. Now $(1,0)\in H$ is not a square.