If every element of $H$ and $G/H$ is a square, then prove that so is every element of $G$. Let $H$ be a subgroup of an abelian group $G$ such that every element of $H$ can be written as $b^2,\, b \in G$ and similarly for $G/H$. Then how to prove that every element of $G$ can also be written as a square ?
Alternative proof. Apply the Snake Lemma to
$$\begin{array}{c} 0 &amp; \rightarrow &amp; H &amp; \rightarrow &amp; G &amp; \rightarrow &amp; G/H &amp; \rightarrow &amp; 0  \\ &amp; &amp; 2 \downarrow ~~&amp; &amp; 2 \downarrow ~~ &amp; &amp; 2 \downarrow ~~ \\0 &amp; \rightarrow &amp; H &amp; \rightarrow &amp; G &amp; \rightarrow &amp; G/H &amp; \rightarrow &amp; 0 \end{array}$$
Hint: Note that $gH=aHaH=a^2H$ so elements of the coset $gH$ have the form...what?