Infinitely presented simple groups separated by homological finiteness properties

Given a finitely generated linear group G over Q, we construct a simple group $Γ$ that has the same finiteness properties as G and admits G as a quasi-retract. As an application, we construct a simple group of type FP∞ that is not finitely presented. Moreover we show that for every n ∈ N there is a simple group of type FPₙ that is neither finitely presented nor of type FPₙ₊₁. Since our simple groups arise as Röver--Nekrashevych groups, this answers a question of Zaremsky.