Haar measure and push forward in the subset of squares

Question

Let $K$ be a compact group with Haar probability measure $m$, let $K^2 = \{k^2 : k\in K\}$ and suppose that $m(K^2)>0$. Show that if $m(A\cap K^2) = m(K^2)$ then $m(\{k\in K : k^2\in A\})=1$.