A question about the definition of PRFs
Let $F_n$ be a random involution induced by a random perfect matching on $\{0,1\}^n$. For any input $x$, the distributions of $F_n(x)$ and $R_n(x)$ are almost identical (in fact, $F_n(x)$ has the same distribution as $R_n(x)$ conditioned on the event $R_n(x) \neq x$, which occurs with probability $1-2^{-n}$), so they are statistically indistinguishable. However, $H_n$ is the identity function, so it is easy to distinguish from a random function.

With some more work, it might be possible to extend this from one query to any fixed polynomial number of queries.