Let $M$ be a connected closed oriented manifold with at least one orientation-reversing homeomorphism $M\to M$.

Let $S\subset M$ be a connected closed embedded submanifold of lower dimension. Let $f:M\to M$ be an orientation-preserving homeomorphism.

Is there an orientation-reversing homeomorphism $g:M\to M$ such that $f|_S=g|_S$?