Zero-sum partitions of Abelian groups of order $2^n$

The following problem has been known since the 80's. Let $\Gamma$ be an Abelian group of order $m$ (denoted $|\Gamma|=m$), and let $t$ $m_i$, $1 \leq i t$, positive integers such that $\sum_{i=1}^t m_i=m-1$. Determine when $\Gamma^*=\Gamma\setminus\{0\}$, set non-zero elements $\Gamma$, can partitioned into disjoint subsets $S_i$, $|S_i|=m_i$ $\sum_{s\in S_i}s=0$ for every $i$, t$. It is easy to check $m_i\geq 2$ (for t$) $|I(\Gamma)|\neq 1$ are necessary conditions existence partitions, where $I(\Gamma)$ involutions $\Gamma$. was proved condition sufficient if only $|I(\Gamma)|\in\{0,3\}$. For other groups (i.e., which 3$ $|I(\Gamma)|>1$), case any with $\Gamma\cong(Z_2)^n$ some integer $n$ analyzed completely so far, it shown independently by several authors in this case. Moreover, recently Cichacz Tuza that, $|\Gamma|$ large enough $|I(\Gamma)|>1$, then 4$ sufficient. In paper we generalize result $2^n$. Namely, show $|I(\Gamma)|>1$ $|\Gamma|=2^n$, $n$. We also present applications graph magic- anti-magic-type labelings.