We improve methods for studying \(\kappa \) -factor functions and obtain an asymptotic formula \(\sum _{\begin{array}{c} n\le x\\ n\equiv a\,(mod \,q) \end{array}} {f(n)}\) for a general class of multiplicative function \(f\) . This method applies to a general \(\kappa \) -factor function and the negative real moment of the strong restrictive factor function. More generally, it applies to a more general arithmetic function \(f_A(n)\) with \(A \in \textrm{PSL}_2(\mathbb {Z})\) which includes the \(\kappa \) -factor function, the irrational factor functions, and the restrictive factor function. This general approach allows us to unify the study of \(\kappa \) -factor functions, irrational factor functions, and restrictive factor functions into the investigation of a broader class of functions.
