Let $M_{n}$ be the algebra of all $n\times n$ complex matrices and let $A\in M_{n}$. We denote by $A^{\dagger }$, $A^{D}$, and $A^{\bigcirc \!\!\! \dagger}$ the Moore-Penrose inverse, the Drazin inverse, and the core-EP inverse of $A$, respectively. The weak group inverse $A^{\bigcirc \!\!\!\! w}$ of $A$ may be defined as $A^{\bigcirc \!\!\!\! w}=( A^{\bigcirc \!\!\! \dagger}) ^{2}A$, and the weak core part of $A$ is the matrix $C=AA^{\bigcirc \!\!\!\! w}A$. \par In the paper, the authors introduce the weak group-star matrix of $A\in M_{n}$ (or the weak group-star inverse of $( A^{\dagger }) ^{\ast }$) as the unique solution of the following system of equations: $$ X( A^{\dagger }) ^{\ast }X=X,\quad AX=CA^{\ast },\quad X( A^{\dagger }) ^{\ast }=A^{D}C, $$ where $C$ is the weak core part of A. They denote the weak group-star matrix of $A$ by $A^{{\bigcirc \!\!\!\! w},\ast }$. \par The paper has seven sections. After introducing the weak group-star matrix of $A$, the authors prove in Section 2 that $A^{{\bigcirc \!\!\!\! w},\ast }=A^{\bigcirc \!\!\!\! w}AA^{\ast }$ and compare $A^{{\bigcirc \!\!\!\! w},\ast }$ with some well-known generalized inverses of $A$. They also present several characterizations of this type of matrix, and introduce a new relation as follows. For $A,B\in M_{n}$, we say $A$ is below $B$ under the relation $ \leq ^{{\bigcirc \!\!\!\! w},\ast }$ if $$ AA^{{\bigcirc \!\!\!\! w},\ast }=BA^{{\bigcirc \!\!\!\! w},\ast }\quad \text{and}\quad AA^{{\bigcirc \!\!\!\! w},\ast }A=A^{{\bigcirc \!\!\!\! w},\ast }B. $$ With an example, the authors show that this relation is not a partial order since it is not antisymmetric. In Section 3, a successive matrix squaring computational iterative scheme is given for calculating the weak group-star matrix, and $A^{{\bigcirc \!\!\!\! w},\ast }$ is then calculated for a given $3\times 3$ matrix $A$. Cramer's rule for the solution of the equation $( A^{\dagger }) ^{\ast }x=b$ is presented in Section 4. In Section 5, the authors use the core-EP decomposition to study the perturbation of $A^{{\bigcirc \!\!\!\! w},\ast }$. The weak group-star matrix is used in Section 6 for solving certain systems of linear equations. The authors conclude the paper by summarizing Sections 2--6 and introducing the dual version $A^{\ast ,{\bigcirc \!\!\!\! w}}$ of the weak group-star matrix of $A\in M_{n}$.