Let $X = BG \times [0; 1]$, then $\ast_a = (\ast; 0)$, $\ast_b = (\ast; 1)$, $\gamma$ is the image of $[0;1]$, $f_a$ is any choice of path in the class of $a$ and $f_b$ is any choice of path in class $b$ conjugated by $\gamma$, i.e. $\gamma \bullet b \bullet \gamma^{-1}$.

Let $X = BG \times [0; 1]$, then $\ast_a = (\ast; 0)$, $\ast_b = (\ast; 1)$, $\gamma$ is the image of $[0;1]$, $f_a$ is any choice of path in the class of $a$ and $f_b$ is any choice of path in class $b$ conjugated by $\gamma$, i.e. $\gamma \bullet b \bullet \gamma^{-1}$.

Equivalently, this is the action groupoid for the trivial action of $G$ on the unit interval $[0;1]$.