We investigate a $H$-invariant linear code $C$ over the finite field $F_p$ where $H$ is a group of linear transformations. We show that if $H$ is a noncyclic abelian group and $\left(\right|H|,p)=1$, then the code $C$ is the sum of the centralizer codes $C_c\left(h\right)$ where $h$ is a nonidentity element of $H$. Moreover if $A$ is subgroup of $H$ such that $A\cong Z_q\times Z_q$, $q\ne p$, then dim $C$ is known when the dimension of $C_c\left(K\right)$ is known for each subgroup $K\ne 1$ of $A$. In the last few sections we restrict our scope of investigation to a special class of invariant codes, namely affine...