  The classical Poisson functor associates to every infinite measure preserving
dynamical system $(X,\mu,T)$ a probability preserving dynamical system
$(X^*,\mu^*,T_*)$ called the Poisson suspension of $T$. In this paper we
generalize this construction: a subgroup Aut$_2(X,\mu)$ of $\mu$-nonsingular
transformations $T$ of $X$ is specified as the largest subgroup for which $T_*$
is $\mu^*$-nonsingular. Topological structure of this subgroup is studied. We
show that a generic element in Aut$_2(X,\mu)$ is ergodic and of Krieger type
III$_1$. Let $G$ be a locally compact Polish group and let
$A:G\to\text{Aut}_2(X,\mu)$ be a $G$-action. We investigate dynamical
properties of the Poisson suspension $A_*$ of $A$ in terms of an affine
representation of $G$ associated naturally with $A$. It is shown that $G$ has
property (T) if and only if each nonsingular Poisson $G$-action admits an
absolutely continuous invariant probability. If $G$ does not have property
$(T)$ then for each generating probability $\kappa$ on $G$ and $t>0$, a
nonsingular Poisson $G$-action is constructed whose Furstenberg
$\kappa$-entropy is $t$.
