Is a subgroup $H$ of a group $G$ normal if $g^2 \in H$ for all $g \in G$?

Just think about this: $$g^{-2}h^{-1}(hg)^2\in H$$ wherein $g\in G$ is any element and $h\in H$.

Here is a different solution.

Let $N$ be the subgroup generated by the elements $g^2$, where $g \in G$. Then $N$ is a normal subgroup, and $G/N$ is abelian since $x^2 = 1$ for each $x \in G/N$. So if $g^2 \in H$ for all $g \in G$, it follows that $H$ contains $N$. Because $G/N$ is abelian, $H/N$ is a normal subgroup of $G/N$ and thus $H$ is a normal subgroup of $G$.