What is the probability that composing two random derangements yields another derangement?

A slightly more specific question: suppose we start with a given derangement and compose it with all derangements. What is the distribution of cycle structures?

Addendum: perhaps it would be worth explaining why this question popped up in the first place. I am interested (purely out of curiosity) in the expected number of "steps" it takes to reach the identity permutation where each step is composition with a random derangement. See here for a related question (the only different is starting with a random permutation).

If you like to have "context": this is like asking how many steps it takes for $n$ penguins/koalas/beings with terrible memory to find their room, which they will recognize upon being inside, but which is otherwise indistinguishable. That's why I originally began with an arbitrary permutation, in any case.