Most heads when coins can only be turned in pairs

You toss four fair coins. Afterwards you may turn coins over (heads to tails or tails to heads), but only two at a time; flipping a lone coin is not allowed. You may do this as many times as you wish. With the best play, what is the expected number of heads at the end?

$7/2$. Turning two coins preserves the parity of the number of heads, and any configuration of the same parity is reachable. An even head count can become $4$ heads; an odd one can reach at most $3$. Each parity has probability $\tfrac12$, so the answer is $\tfrac{4+3}{2}$.