struct SpecialAutomorphismGroup{N} <: SymmetrizedGroup
    group::AutGroup
end

function SpecialAutomorphismGroup(args::Dict)
    N = args["SAut"]
    return SpecialAutomorphismGroup{N}(AutGroup(FreeGroup(N), special=true))
end

name(G::SpecialAutomorphismGroup{N}) where N = "SAutF$(N)"

group(G::SpecialAutomorphismGroup) = G.group

function generatingset(G::SpecialAutomorphismGroup)
    S = gens(group(G));
    return unique([S; inv.(S)])
end

function autS(G::SpecialAutomorphismGroup{N}) where N
    return WreathProduct(PermutationGroup(2), PermutationGroup(N))
end

###############################################################################
#
#  Action of WreathProductElems on AutGroupElem
#
###############################################################################

function AutFG_emb(A::AutGroup, g::WreathProductElem)
    isa(A.objectGroup, FreeGroup) || throw("Not an Aut(Fₙ)")
    parent(g).P.n == length(A.objectGroup.gens) || throw("No natural embedding of $(parent(g)) into $A")
    elt = A()
    Id = parent(g.n.elts[1])()
    flips = Groups.AutSymbol[Groups.flip_autsymbol(i) for i in 1:length(g.p.d) if g.n.elts[i] != Id]
    Groups.r_multiply!(elt, flips, reduced=false)
    Groups.r_multiply!(elt, [Groups.perm_autsymbol(g.p)])
    return elt
end

function AutFG_emb(A::AutGroup, p::perm)
    isa(A.objectGroup, FreeGroup) || throw("Not an Aut(Fₙ)")
    parent(p).n == length(A.objectGroup.gens) || throw("No natural embedding of $(parent(g)) into $A")
    return A(Groups.perm_autsymbol(p))
end

function (g::WreathProductElem)(a::Groups.Automorphism)
    A = parent(a)
    g = AutFG_emb(A,g)
    res = A()
    Groups.r_multiply!(res, g.symbols, reduced=false)
    Groups.r_multiply!(res, a.symbols, reduced=false)
    Groups.r_multiply!(res, [inv(s) for s in reverse!(g.symbols)])
    return res
end

function (p::perm)(a::Groups.Automorphism)
    g = AutFG_emb(parent(a),p)
    return g*a*inv(g)
end