In this paper, we study the problem of removing an element from an additive basis in a general abelian group. We introduce analogues of the classical functions $X$, $S$ and $E$ (defined in the case of the integers) and obtain bounds on them. Our estimates on the functions $S_G$ and $E_G$ are valid for general abelian groups $G$ while in the case of $X_G$ we show that distinct types of behaviours may occur depending on the group $G$.