Schur-triviality is not quotient-closed

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This article gives the statement, and possibly proof, of a group property (i.e., Schur-trivial group) not satisfying a group metaproperty (i.e., quotient-closed group property).
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Statement

It is possible to have a Schur-trivial group G (i.e., the Schur multiplier of G is the trivial group) and a normal subgroup H of G such that the quotient group G/H is not Schur-trivial.

Proof

Example of the quaternion group

This is the smallest example:

Perfect group examples

We can construct many examples this way: let G be the universal central extension of some perfect group M that is not Schur-trivial. Then, G is a Schur-trivial group, and M is the quotient of G by some central subgroup, and is not Schur-trivial.