Automorphism group of disjoint cycle graphs of different lengths

Question

This question is supplementary to another question.

From that question, we know that the automorphism group of the $N$ disjoint cycle graphs of same length $n$ is $S_N \wr D_n$.

My question: What is the Automorphism group of disjoint cycle graphs of different lengths?

My effort: I understand that there will be two cases:

1. More than one cycle graphs with same length
2. Only one cycle graph for each length

Can anyone help me to find for a general solution or special solution to each of these cases?