Let $X$ be a first countable Hausdorff topological group. The limit of a sequence in $X$ defines a function denoted by lim from the set of all convergent sequences to $X$. This notion has been modified by Connor and Grosse-Erdmann for real functions by replacing lim with an arbitrary linear functional $G$ defined on a linear subspace of the vector space of all real sequences. Recently \c Cakall\i has extended the concept to the topological group setting and introduced the concepts of $G$-sequential compactness, $G$-sequential continuity and sequential connectedness. In this paper we give a further investigation of $G$-sequential continuity in topological groups.