Is an automorphism of a finite group inner when it preserves conjugacy of elements and subgroups?

The script below is a GAP-script that should identify the existence of such automorphism $\varphi$ for a given group $G$. Note that I've done this in a hurry, so I can't guarantee the correctness.

SatisfiesCriteria := function(G)
    local A,phi,Conj,Subs,C,H;
    A := AutomorphismGroup(G);
    Conj := ConjugacyClasses(G);
    Subs := ConjugacyClassesSubgroups(G);
    for phi in A do
        # Is phi not inner?
        if IsInnerAutomorphism(phi) then
            continue;
        fi;

        # Is phi class-preserving?
        if ForAny(Conj, C -> not Image(phi,Representative(C)) in C) then
            continue;
        fi;

        # Is phi subgroup-class-preserving?
        if ForAny(Subs, H -> not Image(phi,Representative(H)) in H) then
            continue;
        fi;
        return phi;
    od;
    return fail;
end;

The script seems to find such automorphism for SmallGroup(32,44). In particular, this group is isomorphic to the finitely presented group

<fp group of size 32 on the generators [ F1, F2, F3, F4, F5 ]>

with relators

[ F1^2, F2^-1*F1^-1*F2*F1*F4^-1, F3^-1*F1^-1*F3*F1*F5^-1, 
  F4^-1*F1^-1*F4*F1*F5^-1, F5^-1*F1^-1*F5*F1, F2^2*F5^-1, F3^-1*F2^-1*F3*F2, 
  F4^-1*F2^-1*F4*F2*F5^-1, F5^-1*F2^-1*F5*F2, F3^2, F4^-1*F3^-1*F4*F3, 
  F5^-1*F3^-1*F5*F3, F4^2*F5^-1, F5^-1*F4^-1*F5*F4, F5^2 ]

and the automorphism is given by

[ F1, F2, F3, F4, F5 ] -> [ F1, F2*F4, F3, F4*F5, F5 ]

Note: the example from GroupProps you mention is SmallGroup(32,43) in GAP. If we leave out the condition of being subgroup-class-preserving, then the above script indeed finds a non-inner automorphism that is class-preserving, though I haven't checked if it's the same one.

Note 2: In On groups with a class-preserving outer automorphism by Brooksbank and Mizuhara, the following is said:

In 1947, Wall showed that, for each integer $m$ divisible by 8, the general linear group $\operatorname{GL}(1,\mathbb{Z}/m)$, of order $m \cdot \varphi(m)$ has a nearly inner automorphism that is not inner inner [Wa]. These include a smallest example of such groups, namely $\operatorname{GL}(1,\mathbb{Z}/8)$ of order $32$. (There are actually two non-isomorphic groups of order $32$ having this property.)

These two non-isomorphic groups would then be SmallGroup(32,43) and SmallGroup(32,44).