Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 104 / 6 / d / ii

Part (c) shows that every finitely presented group embeds in a finitely presented group with no nontrivial finite quotients of a group. Consequently this property also has no possible forbidden witness $A_{-}$. ThereforeBoth properties have positive witnesses, for example $C_2$ and the trivial group respectively. It is the required embedding obstruction that fails in each case.