an algebra question about hermitian and unitary

an algebra question about hermitian and unitary

If $H_1$ and $H_2$ are two hermitian matrices, $U_1$ and $U_2$ are two
unitary matrices ($H_1\ne U_1$, $H_2\ne U_2$), $A$ is a square matrix,
then if $H_1AH_2=U_1AU_2$, is it true that $H_1AH_2=A$?