Fundamental group after attaching a Möbius band

A Möbius band retracts to its core circle, and its boundary wraps twice around the core. Attaching it to $X$ along a loop representing $w\in {\pi }_1(X)$ therefore adjoins a generator $x$ with relation $x^2=w$.

Attach Möbius bands to a torus along the classes $ab$ and $a^2{b}^3$. If their core generators are $x,y$, the resulting presentation isBecause the exponent vectors $(1,1)$ and $(2,3)$ form a unimodular basis of ${\mathbb{Z}}^2$, eliminating $a,b$ givesThis group maps onto $S_3$ by $x\mapsto (12)$ and $y\mapsto (23)$, so it is nonabelian.