function (p::perm)(A::GroupRingElem)
    RG = parent(A)
    result = zero(RG, eltype(A.coeffs))
    
    for (idx, c) in enumerate(A.coeffs)
        if c!= zero(eltype(A.coeffs))
            result[p(RG.basis[idx])] = c
        end
    end
    return result
end

###############################################################################
#
#  Action of WreathProductElems on Nemo.MatElem
#
###############################################################################

function matrix_emb(n::DirectProductGroupElem, p::perm)
    Id = parent(n.elts[1])()
    elt = diagm([(-1)^(el == Id ? 0 : 1) for el in n.elts])
    return elt[:, p.d]
end

function (g::WreathProductElem)(A::MatElem)
    g_inv = inv(g)
    G = matrix_emb(g.n, g_inv.p)
    G_inv = matrix_emb(g_inv.n, g.p)
    M = parent(A)
    return M(G)*A*M(G_inv)
end

import Base.*

doc"""
    *(x::AbstractAlgebra.MatElem, P::Generic.perm)
> Apply the pemutation $P$ to the rows of the matrix $x$ and return the result.
"""
function *(x::AbstractAlgebra.MatElem, P::Generic.perm)
   z = similar(x)
   m = rows(x)
   n = cols(x)
   for i = 1:m
      for j = 1:n
         z[i, j] = x[i,P[j]]
      end
   end
   return z
end

function (p::perm)(A::MatElem)
    length(p.d) == A.r == A.c || throw("Can't act via $p on matrix of size ($(A.r), $(A.c))")
    return p*A*inv(p)
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(p)) 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