Absolute continuity of the push forward of the Haar measure

Question

Let $K$ be a compact group with Haar measure $m$. Let $K^2 = \{k^2 : k\in K\}$ and suppose that $K = K^2 \cup wK^2$ for some $w\in K$. Let $\mu$ be the push-forward of $m$ under the map $k\mapsto k^2$. Is it true that $\mu \ll m|_{K^2} \ll \mu$?

Answer 1

No, take $K=O(2).$ Then $m(K^2)=m(SO(2))=\tfrac12$ but $\mu(\{1\})=\tfrac12,$ so $\mu\not \ll m.$