Let $P$ be a random permutation of $n>1$ bits. Let $F$ be the function on the same domain $\{0,1\}^n$, defined by $F(x)=P(x)\oplus x$. When $P$ is a block cipher with key a message block, that's the Davies-Meyer construction of a one-way compression function, a variant of which being used in MD5, SHA-1, and SHA-2.
Let $H_F$ be the entropy (in bits) of the source $F(x)$ where $x$ is a vector of $n$ random unbiased independent bits. That is $$\begin{align*} H_F&=\sum_{y\in F(\{0,1\}^n)}-\Pr(F(x)=y)\cdot\log_2\big(\Pr(F(x)=y)\big)\\ \\ & = n-{1\over2^n}\cdot\sum_{y\in F(\{0,1\}^n)}\big(\#{\{x, F(x)=y)\}}\big)\cdot\log_2\big(\#{\{x, F(x)=y)\}}\big) \\ & = n-{1\over2^n}\cdot\sum_{2\le\,j\,\le2^n}\big(\#{\{y,(\#\{x:F(x)=y\})=j\}}\big)\cdot j\cdot\log_2(j) \end{align*}$$ Note: $\#{\{x, F(x)=y)\}}$ of the 2nd equation is the number of preimages of vector $y$; and $\#{\{y,(\#\{x:F(x)=y\})=j\}}$ of the 3rd is the number of vectors with exactly $j$ preimages.
$H_F$ can be from $0$ (when $P$ happens to be XOR with some constant) to $n$ bits (when $F$ happens to be a permutation, e.g. for $P:$ $00\to00$, $01\to10$, $10\to11$, $11\to01$ ).
Let $H(n)$ be the expected value of $H_F$; that is, the average of $H_F$ over all $(2^n)!$ permutations $P$. A plausible conjecture is that $$\lim_{n\to\infty}{H(n)\over n}=1$$ How to prove this, and what's a first order approximation of $n-H(n)$ for large $n$?
If $F$ was a random function, we'd have $n-H(n)\approx0.8272\dots$ starting with moderate $n$ according to my analysis there. But I fail to rigorously derive that for the much narrower class of $F$ in the question.