Lifting mixing properties by Rokhlin cocycles

We study the problem of lifting various mixing properties from a base automorphism $T\in {\rm Aut}\xbm$ to skew products of the form $\tfs$, where $\va:X\to G$ is a cocycle with values in a locally compact Abelian group $G$, $\cs=(S_g)_{g\in G}$ is a measurable representation of $G$ in ${\rm Aut}\ycn$ and $\tfs$ acts on the product space $(X\times Y,\cb\ot\cc,\mu\ot\nu)$ by $$\tfs(x,y)=(Tx,S_{\va(x)}(y)).$$ It is also shown that whenever $T$ is ergodic (mildly mixing, mixing) but $\tfs$ is not ergodic (is not mildly mixing, not mixing), then on a non-trivial factor $\ca\subset\cc$ of $\cs$ the corresponding Rokhlin cocycle $x\mapsto S_{\va(x)}|_{\ca}$ is a coboundary (a quasi-coboundary).