  Let $\Gamma$ be a countable discrete group, and let $\pi\colon \Gamma\to
{\rm{GL}}(H)$ be a representation of $\Gamma$ by invertible operators on a
separable Hilbert space $H$. We show that the semidirect product group
$G=H\rtimes_{\pi}\Gamma$ is SIN ($G$ admits a two-sided invariant metric
compatible with its topology) and unitarily representable ($G$ embeds into the
unitary group $\mathcal{U}(\ell^2(\mathbb N))$), if and only if $\pi$ is
uniformly bounded, and that $\pi$ is unitarizable if and only if $G$ is of
finite type: that is, $G$ embeds into the unitary group of a II$_1$-factor.
Consequently, we show that a unitarily representable Polish SIN groups need not
be of finite type, answering a question of Sorin Popa. The key point in our
argument is an equivariant version of the Maurey--Nikishin factorization
theorem for continuous maps from a Hilbert space to the space $L^0(X,m)$ of all
measurable maps on a probability space.
