Let $A$ be a $C^*$-algebra and let $\phi:A\to B(H)$ be a completely positive map. The Stinespring representation theorem constructs a representation of $A$ on a Hilbert space $K$, which is constructed as follows. Define $K_0=A\otimes H$ with the inner product $$\langle a\otimes v, b\otimes w\rangle_{\phi}= \langle \phi(b^*a)v,w\rangle_H,$$ extended linearly. Then $K$ is the closure of $K_0/\{z:\langle z,z\rangle_\phi=0 \}$ in the norm induced by the inner product $\langle \cdot,\cdot\rangle_\phi$.

Given a unitary linear operator $T$ on $H$ we can consider an operator $S$ on $K$ induced by $$S(a\otimes v)=a\otimes (Tv).$$

My question is under what conditions on $T$ is $S$ well-defined and bounded?

In general, I don't want to assume that $T$ commutes with the range of $\phi$.