Let me begin by mumbling some abstract nonsense, and then attempt to be concrete. The category of groups inherits the structure of a strict 2-category from the 2-category of small categories. Explicitly, a 2-morphism between $\varphi$ and $\varphi'\colon H \to G$ is an element $g \in G$ such that for each $h \in H$, we have $\varphi(h) = g\varphi(h)g^{-1}$. I'm curious why this category has coinserters but not inserters.

So much for the mumbling.
Let $\varphi$ and $\varphi'\colon H \to G$ be homomorphisms of groups. Let $\psi\colon K \to H$ be a homomorphism and $g$ an element of $G$. We say that the pair $(\psi,g)$ *inserts* $\varphi$ and $\varphi'$ if we have that for all $k \in K$, the following equality holds in $G$

$$\varphi\psi(k) = g\varphi'\psi(k)g^{-1}.$$

The collection of pairs $(\psi,g)$ with domain $K$ that inserts $\varphi$ and $\varphi'$ forms a *category*: a morphism in this category from $(\psi,g)$ to $(\psi',g')$ is an element $h \in H$ such that for all $k$ in $K$, we have the equality

$$\psi(k) = h\psi'(k)h^{-1}$$

as well as the equality

$$\varphi(h)g' = g\varphi'(h).$$

A *strict inserter* of $\varphi$ and $\varphi'$ would be a group $\mathrm{Ins}(\varphi,\varphi')$ equipped with an inserting pair $(\Psi,g)$ which is *universal* for this property in the following sense:
If $(\psi',g')$ is an inserting pair with domain $K$, there exists a unique homomorphism $\eta\colon K \to \mathrm{Ins}(\varphi,\varphi')$ such that
$\psi' = \Psi\eta$ and such that $g = g'$. This latter requirement kills the possibility of strict inserters: even if $G$ is *abelian,* so that the condition of a pair $(\psi,g)$ inserting $\varphi$ and $\varphi'$ is just the condition that $\psi$ equalizes $\varphi$ and $\varphi'$, the free choice of $g$ makes it impossible to have a universal choice of $g$.

The dual notion is of a *coinserter*. This is a group $K$ equipped with a homomorphism $\Psi\colon G \to K$ and an element $t \in K$ such that $\Psi\varphi(h) = t\Psi\varphi'(h)t^{-1}$ for all $h \in H$. If a pair $(\psi',t')\colon G \to K'$ coinserts $\varphi$ and $\varphi'$, there must be a unique homomorphism $K \to K'$ satisfying the obviously dual condition to the above.
Now since $t$ and $t'$ live in *different groups,* suddenly we're able to construct coinserters, certainly at least in the case where $\varphi$ and $\varphi'$ are injective, but I imagine a variant of the construction produces a coinserter in all cases.

Explicitly the coinserter is the pair $(\iota, t)\colon G \to G*_H$,
where $G*_H$ denotes the *HNN extension* of $G$ with associated subgroups $\varphi(H)$ and $\varphi'(H)$, $\iota$ is the canonical inclusion of $G$ into $G*_H$ and $t$ is the "stable letter" for the HNN extension.

A presentation for $G*_H$, given that a presentation for $G$ is $G = \langle S \mid R \rangle$ is as follows:

$$\langle S, t \mid R, t\varphi(h)t^{-1} = \varphi'(h) \rangle$$

as $h \in H$ varies. It should be clear that if a pair $(\psi,t')\colon G \to K$ coinserts $\varphi$ and $\varphi'$, we can define a homomorphism $G*_H \to K$ taking $\iota(g)$ to $\psi(g)$ for all $g \in G$ and taking $t$ to $t'$.

Sorry for the ramble. Anyway: is there some deeper reason that the 2-category of groups should have (strict) coinserters but not (strict) inserters? After all, the 2-category of categories has both.