Conjugacy testing problem

The below-given problem is in black box setting means input is given by set of generators.

Given an abelian $$p$$-group $$A$$ and two matrices $$U_1$$ and $$U_2$$ in $$R(A)$$ such that the order of $$U_1$$ and $$U_2$$ are coprime with $$p$$, output an element $$U \in R(A)$$ such that $$UU_1=U_2U$$ if such an element exists.

$$R(A)$$ denotes the set of all automorphisms of $$A$$.

The brute force way seems to give a $$\Theta(|A|^{\log |A|})$$ bound.

Question: Is the above problem solvable in the polynomial time?