Compactness of a special kind of Integral operators

Question

Let $(S(t))_{t>0}$ be a continuous operator from $L^2(0,1)$ to its self and Let $K$ be the operator $$\eqalign{ & K:{L^2}(0,1) \to {L^2}(0,1) \cr & f: \to (Kf)(x) = \int\limits_0^1 {k(s,x)S(s)f(s)ds} \cr} $$ where $$k \in {L^2}(0,1) \times {L^2}(0,1)$$ It is well known that if $S=I$ then K is compact operator. What can I say about the compactness of $K$ is this case? Thank you.