Let $S$ be a finite group with operator + and $\pi$ be a permutation on $S$. Then what is the probability that $\pi(x) + x$ is injective over choices of $\pi$?
The concrete instantiation I'm interested in is $S=$GF$(2^n)$ for fixed $n > 0$. (Computer Scientists call this "xor on $n$-bit strings.")
For $n=1$ we have two permutations, neither of which produce an injection. For $n=2$ we have 24 permutations, 8 of which induce an injection so the probability is 1/3.
Here is an example $\pi$ for $n=2$:
$$\pi(0)=0,\ \pi(1) = z,\ \pi(z)=z+1,\ \pi(z+1)=1.$$
Here notice that $\pi(x) + x$ produces $0$, $z+1$, $1$, and $z$, respectively, which is an injection.