  We obtain a complete characterization of the entire functions $g$ such that
the integral operator $(T_ g f)(z)=\int_{0}^{z}f(\zeta)\,g'(\zeta)\,d\zeta$ is
bounded or compact, on a large class of Fock spaces $\mathcal{F}^\phi_p$,
induced by smooth radial weights that decay faster than the classical Gaussian
one. In some respects, these spaces turn out to be significantly different than
the classical Fock spaces. Descriptions of Schatten class integral operators
are also provided. En route, we prove a Littlewood-Paley formula for
$||\cdot||_{\mathcal{F}^\phi_p}$ and we characterize the positive Borel
measures for which $\mathcal{F}^\phi_p\subset L^q(\mu)$, $0<p,q<\infty$. In
addition, we also address the question of describing the subspaces of
$\mathcal{F}^\phi_p$ that are invariant under the classical Volterra integral
operator.
