I'm trying to find all finite intransitive permutation 2-groups $G$ with the following property: all of their elements in their disjoint cycle decompositions have cycles of equal lengths, i.e. if $g\in G$ and $g=l_1 l_2 \cdots l_s$, then $|l_1|=|l_2|=\cdots=|l_s|=l$. Obviously, elementary abelian 2-groups are examples. Can we prove that only elementary abelian 2-groups and cyclic 2-groups have this property?
Edition: As Professor Derek Holt said cyclic groups are other examples.