Summary: A continuous semigroup of probability generating functions \(F\) is used to introduce a notion of semi-selfdecomposability, called \(F\)-semi-selfdecomposability, for distributions with support on the nonnegative integers. \(F\)-semi-selfdecomposable distributions are infinitely divisible and are characterized by the absolute monotonicity of a specific function. The class of \(F\)-semi-selfdecomposable laws is shown to contain the \(F\)-semistable distributions and the geometric \(F\)-semistable distributions. A generalization of discrete random stability is also explored.