  Let $p$ be a prime and let $\pi^n(X;\mathbb{Z}/p^r)=[X,M_n(\mathbb{Z}/p^r)]$
be the set of homotopy classes of based maps from CW-complexes $X$ into the mod
$p^r$ Moore spaces $M_n(\mathbb{Z}/p^r)$ of degree $n$, where $\mathbb{Z}/p^r$
denotes the integers mod $p^r$. In this paper we firstly determine the modular
cohomotopy groups $\pi^n(X;\mathbb{Z}/p^r)$ up to extensions by classical
methods of primary cohomology operations and give conditions for the splitness
of the extensions. Secondly we utilize some unstable homotopy theory of Moore
spaces to study the modular cohomotopy groups; especially, the group
$\pi^3(X;\mathbb{Z}_{(2)})$ with $\dim(X)\leq 6$ is determined.
