Unique ergodicity a spectral invariant?

Question

Suppose $f:X\to X$ and $g:Y\to Y$ are two automorphism of measure spaces. Suppose there is an unitary operator $V:L^2(X)\to L^2(Y)$ satisfying $VU_fV^*=U_g$, here $U_f$ and $U_g$ are the Koopman operators corresponding to $f$ and $g$. Additionally we know $f$ is uniquely ergodic. Can we say $g$ is uniquely ergodic? I suspect no. Counterexample?