In teaching my algebraic topology class, this group showed up as part of an easy fundamental group computation: $\langle a,b\mid a^2=b^2\rangle$. My first instinct was that this must be $\mathbb{Z}*\mathbb{Z}/2$ because clearly every element can be written as a product of $b$'s (only to the power 1) and powers of $a$. But this turns out to be far from clear (and likely wrong). I assume this must be a well-known group to group theorists, so I'm curious if it's isomorphic to something that can be described by other means (or what's known about it in general). 

Thanks!

Setting $c:=b^{-1}$ one obtains the presentation
$$G= \langle a,c \,  | \, a^2c^2=1 \rangle,$$
which is the fundamental group of the Klein bottle.

It is well known that another presentation of such a group is 
$$G= \langle x,y \,|\, x^{-1}yx=y^{-1} \rangle,$$
and this allows one to write $G$ as a semi-direct product of infinite cyclic groups, namely $$G= \mathbb{Z} \rtimes_{\sigma} \mathbb{Z}$$
where $\sigma \colon \mathbb{Z} \to \mathbb{Z}$ is defined by $\sigma(y)=y^{-1}$ and $\mathbb{Z}$ is written multiplicatively.
