Characteristic subgroup growth

Let snch(Γ) denote the number of characteristic subgroups of index at most n in a finitely generated group Γ. In response to a question of I. Rivin, we show that if Γ=Fr is the free group on r≥2 generators, then the growth type of snch(Fr) is nlog(n). This is in contrast with the expectation of W. Thurston who predicted that there should be a difference between r=2 and r>2. Along the way, we answer a question of Barnea and Schlage-Puchta (J Group Theory 23(1):1–15, 2020) on the normal subgroup growth of large groups.