We first present a new form of a local $K$-convoluted $C$-group on a Banach space $X$, and then deduce some basic properties of a nondegenerate local $K$-convoluted $C$-group on $X$ and some generation theorems of local $K$-convoluted $C$-groups, which can be applied to obtain some equivalence relations between the generation of a nondegenerate local $K$-convoluted $C$-group on $X$ with subgenerator $A$ and the unique existence of solutions of the abstract Cauchy problem $\text{ACP}(A,f,x)$.