A complete mapping of a group $Γ$ is a bijection $\varphi\colon Γ\to Γ$ for which the mapping $x \mapsto x+\varphi(x)$ is a bijection. In this paper we consider the existence of a complete mapping $\varphi$ of $Γ$ and a partition $S_1,S_2,\ldots S_t$ of elements of $Γ$, such that $\sum_{s\in S_i}s=\sum_{s\in S_i}\varphi(s)=0$ for every $i$, $1 \leq i \leq t$. A $Γ$-magic rectangle set $MRS_Γ(a, b; c)$ of order $abc$ is a collection of $c$ arrays $(a\times b)$ whose entries are elements of group $Γ$ of order $abc$, each appearing once, with all row sums in every rectangle equal to a constant $ω\in Γ$ and all column sums in every rectangle equal to a constant $δ\in Γ$. While a complete characterization of MRS$_Γ(a,b;c)$ exists for cases where $\{a,b\}\not=\{2k+1,2^α\}$, the scenario where $\{a,b\}=\{2k+1,2^α\}$ remains unsolved for $α>1$. Using the partition of $Γ$ into zero-sum sets by complete mappings, we give some sufficient conditions that a $Γ$-magic rectangle set MRS$_Γ(2k+1, 2^α;c)$ exists.