= Solution
Every finitely presented <group> $P$ embeds in $P\times C_2$ by $p\mapsto(p,1)$. This <direct product of groups> is finitely presented: combine finite presentations of the factors and add the finitely many cross-commutation relations between their generators. Projection onto $C_2$ gives a nontrivial <finite quotient of a group>.
Thus no finitely presented $P$ can serve as the forbidden witness $A_-$ for this property. Hence
$$
\boxed{\text{Having a nontrivial finite quotient is not a Markov property}.}
$$
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