The paper under review is concerned with combinatorial properties of semidirect products such as finite generation and finite presentability. Let \((Y,\wedge)\) be a semilattice and \((G,.)\) be a group acting on \(Y\). The authors give necessary and sufficient conditions under which \(S=Y\rtimes G\) is finitely generated, in terms of finite generations of the group \(G\) and the semilattice \(Y\) and a condition regarding maximal elements of \(Y\). They then consider finite presentability of this semidirect product and prove that \(S=Y\rtimes G\) is finitely presented as an inverse semigroup if and only if \(G\) is finitely presented, \(Y\) is finitely presented as an inverse semigroup with respect to the action of \(G\) and \(Y\) satisfies the maximum condition. The authors finish their paper by giving two examples to illustrate their main results.