Group with unique Schur covering group

Statement
A group with unique Schur covering group is a group $$G$$ satisfying the following equivalent conditions:


 * 1) It has a unique defining ingredient::Schur covering group, i.e., given Schur covering groups $$K_1,K_2$$ of $$G$$ with covering maps $$\pi_1: K_1 \to G$$ and $$\pi_2: K_2 \to G$$, there is an isomorphism $$\theta:K_1 \to K_2$$ such that $$\pi_1 = \pi_2 \circ \theta$$.
 * 2) $$\operatorname{Ext}^1(G^{\operatorname{ab}},M(G)) = 0$$ where $$G^{\operatorname{ab}}$$ is the defining ingredient::abelianization of $$G$$ (and also identified with $$H_1(G;\mathbb{Z})$$) while $$M(G)$$ is the defining ingredient::Schur multiplier of $$G$$ (and also identified with $$H_2(G;\mathbb{Z})$$). Note that for $$G$$ a finite group, this is equivalent to verifying that $$G^{\operatorname{ab}}$$ and $$M(G)$$ have relatively prime orders.

Examples
The smallest example of a group $$G$$ where neither the abelianization nor the Schur multiplier is trivial, but the Schur covering group is still unique, is alternating group:A4, whose unique Schur multiplier is special linear group:SL(2,3).

In addition, we can take perfect groups and Schur-trivial groups as examples.

Non-examples
The smallest non-example is the Klein four-group, which has two Schur covering groups (dihedral group:D8 and quaternion group). Note that in general the number of Schur covering groups is $$|\operatorname{Ext}^1(G^{\operatorname{ab}},M(G))|$$.