Totally variant subgroups of simple groups

[Arturo Magidin]

Let's call a proper subgroup H of a group G "totally variant" if the
only automorphism of G which fixes H pointwise is the identity
of G (equivalently, if an automorphism of G induces the identity
on H, then it is the identity of G). For example, A_4 is totally
variant in A_5 (and in general, A_n is totally variant in A_{n+1} for
n>4); or M_{10} is totally variant in M_{11}.

Some groups do not have totally variant proper subgroups.
For example, the prime order cycles, or products of cycles of
prime order; or the quaternion group of order 8.

My question, given the fact above about A_n, is the following:

Does every finite nonabelian simple group have a totally variant
subgroup?

I would appreciate any thoughts/references/comments/counterexamples.
Thanks in advance.

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"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes")
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Arturo Magidin
mag...@uclink.berkeley.edu
mag...@math.berkeley.edu