Nonabelian $1$-cohomologies and conjugacy of finite subgroups in some extensions

We describe the conjugacy classes of finite subgroups in some split extensions using the notion of $1$-cocycle and $1$-coboundary with values in a noncommutative group. We prove that each finite subgroup in the automorphism group of a free Lie algebra of rank $3$ is conjugated with a subgroup of the linear automorphism group provided that the group order does not divide the characteristic of the ground field.