Pushout of a subgroup Let $G$ be a group, $H\subseteq G$ be a subgroup and $\iota: H\to G$ the inclusion map. What is the "pushout along $\iota$? How is it constructed?
 A: If I understand you correctly, you want the pushout of this diagram:
$$\require{AMScd}
\begin{CD}
H @>{\iota}>> G \\
@V{\iota}VV \\
G
\end{CD}$$
It's possible to describe it "explicitly": it is the quotient of the free product $G * G$ by the normal subgroup $N$ generated by the relations $\iota_1(h) \equiv \iota_2(h)$ where $\iota_1$ is the inclusion of $H$ in the first factor of $G*G$ and $\iota_2$ the inclusion in the second factor. It is typically denoted by:
$$G *_H G = G * G / \langle \iota_1(h) \equiv \iota_2(h) \rangle.$$
This is a rather badly behaved group. For example if $G = \mathbb{Z}$, $H = 2\mathbb{Z}$, then the group $G$ you get has a presentation given by $\langle x,y \mid x^2 = y^2 \rangle$. It's not a very nice group. I doubt you can say much more than that.
Two extreme examples: if $H = \{e\}$, then $G *_{\{e\}} G = G * G$ is the free product of $G$ with itself. If $H = G$, then $G *_G G$ is isomorphic to $G$.
A: Pushouts (and colimits in general) can always be constructed from coproducts and coequalizers whenever these are available (see Awodey's text on category theory, for example). So the pushout of an arrow $ H \to K$ along along the inclusion $H \hookrightarrow G$ would be the quotient of the coproduct of $G$ and $K$ by the appropriate equivalence relation. Just draw out the coproduct diagram and you'll see what the coequalizer should be.
I haven't ever personally come across a pushout of groups, so I'm not sure what the usual "interpretation" of this object is, but this is most likely the easiest way it would be constructed.
