Internal twists of $L$-functions. II

A nonlinear twist $F(s;f)$ of a function $F(s)$ from the extended Selberg class $\mathcal S^\sharp $ is called internal if it belongs to $\mathcal S^\sharp $. In a previous paper (2014) we showed that, inside a rather general class of nonlinear twists, the internal twists occur only in very special cases; moreover, we gave a first characterization of such twists. Here we complete our previous work by giving a fully detailed description of such internal twists.