diff --git a/test/1712.07167.jl b/test/1712.07167.jl
index b1bd59e..259836a 100644
--- a/test/1712.07167.jl
+++ b/test/1712.07167.jl
@@ -32,6 +32,18 @@
         # this should be very fast due to warmstarting:
         @test λ ≈ PropertyT.spectral_gap(sett) atol=1e-5
         @test PropertyT.check_property_T(sett) == true
+
+        ##########
+        # Symmetrizing by PermGroup(3):
+
+        sett = PropertyT.Settings("SL($N,Z)", G, S, PermGroup(N), with_SCS(4000, accel=20);
+        upper_bound=0.27, warmstart=true)
+
+        PropertyT.print_summary(sett)
+
+        λ = PropertyT.spectral_gap(sett)
+        @test λ > 0.269999
+        @test PropertyT.interpret_results(sett, λ) == true
     end
 
     @testset "oSL(4,Z)" begin
diff --git a/test/actions.jl b/test/actions.jl
new file mode 100644
index 0000000..fd9ce04
--- /dev/null
+++ b/test/actions.jl
@@ -0,0 +1,104 @@
+@testset "actions on Group[Rings]" begin
+    Eij = PropertyT.EltaryMat
+    ssgs(M::MatAlgebra, i, j) = (S = [Eij(M, i, j), Eij(M, j, i)];
+        S = unique([S; inv.(S)]); S)
+
+    rmul = Groups.rmul_autsymbol
+    lmul = Groups.lmul_autsymbol
+
+    function ssgs(A::AutGroup, i, j)
+        rmuls = [rmul(i,j), rmul(j,i)]
+        lmuls = [lmul(i,j), lmul(j,i)]
+        gen_set = A.([rmuls; lmuls])
+        return unique([gen_set; inv.(gen_set)])
+    end
+
+@testset "actions on SL(3,Z) and its group ring" begin
+    N = 3
+    halfradius = 2
+    M = MatrixAlgebra(zz, N)
+    S = PropertyT.generating_set(M)
+    E_R, sizes = Groups.generate_balls(S, one(M), radius=2halfradius);
+
+    rdict = GroupRings.reverse_dict(E_R)
+    pm = GroupRings.create_pm(E_R, rdict, sizes[halfradius]; twisted=false);
+    RG = GroupRing(M, E_R, rdict, pm)
+
+    @testset "correctness of actions" begin
+        Δ = length(S)*RG(1) - sum(RG(s) for s in S)
+        @test Δ == PropertyT.spLaplacian(RG, S)
+
+        elt = S[5]
+        x = RG(1) - RG(elt)
+        elt2 = E_R[rand(sizes[1]:sizes[2])]
+        y = 2RG(elt2) - RG(elt)
+
+        for G in [PermGroup(N), WreathProduct(PermGroup(2), PermGroup(N))]
+            @test all(g(one(M)) == one(M) for g in G)
+            @test all(rdict[g(m)] <= sizes[1] for g in G for m in S)
+            @test all(g(m)*g(n) == g(m*n) for g in G for m in S for n in S)
+
+            @test all(g(Δ) == Δ for g in G)
+            @test all(g(x) == RG(1) - RG(g(elt)) for g in G)
+
+            @test all(2RG(g(elt2)) - RG(g(elt)) == g(y) for g in G)
+        end
+    end
+
+    @testset "small Laplacians" begin
+        for (i,j) in PropertyT.indexing(N)
+            Sij = ssgs(M, i,j)
+            Δij= PropertyT.spLaplacian(RG, Sij)
+
+            @test all(p(Δij) == PropertyT.spLaplacian(RG, ssgs(M, p[i], p[j])) for p in PermGroup(N))
+
+            @test all(g(Δij) == PropertyT.spLaplacian(RG, ssgs(M, g.p[i], g.p[j])) for g in WreathProduct(PermGroup(2), PermGroup(N)))
+        end
+    end
+end
+
+@testset "actions on SAut(F_3) and its group ring" begin
+    N = 3
+    halfradius = 2
+    M = SAut(FreeGroup(N))
+    S = PropertyT.generating_set(M)
+    E_R, sizes = Groups.generate_balls(S, one(M), radius=2halfradius);
+
+    rdict = GroupRings.reverse_dict(E_R)
+    pm = GroupRings.create_pm(E_R, rdict, sizes[halfradius]; twisted=false);
+    RG = GroupRing(M, E_R, rdict, pm)
+
+
+    @testset "correctness of actions" begin
+
+        Δ = length(S)*RG(1) - sum(RG(s) for s in S)
+        @test Δ == PropertyT.spLaplacian(RG, S)
+
+        elt = S[5]
+        x = RG(1) - RG(elt)
+        elt2 = E_R[rand(sizes[1]:sizes[2])]
+        y = 2RG(elt2) - RG(elt)
+
+        for G in [PermGroup(N), WreathProduct(PermGroup(2), PermGroup(N))]
+            @test all(g(one(M)) == one(M) for g in G)
+            @test all(rdict[g(m)] <= sizes[1] for g in G for m in S)
+            @test all(g(m)*g(n) == g(m*n) for g in G for m in S for n in S)
+
+            @test all(g(Δ) == Δ for g in G)
+            @test all(g(x) == RG(1) - RG(g(elt)) for g in G)
+
+            @test all(2RG(g(elt2)) - RG(g(elt)) == g(y) for g in G)
+        end
+    end
+
+    for (i,j) in PropertyT.indexing(N)
+        Sij = ssgs(M, i,j)
+        Δij= PropertyT.spLaplacian(RG, Sij)
+
+        @test all(p(Δij) == PropertyT.spLaplacian(RG, ssgs(M, p[i], p[j])) for p in PermGroup(N))
+
+        @test all(g(Δij) == PropertyT.spLaplacian(RG, ssgs(M, g.p[i], g.p[j])) for g in WreathProduct(PermGroup(2), PermGroup(N)))
+    end
+end
+
+end
diff --git a/test/runtests.jl b/test/runtests.jl
index 96bb5e0..1a0e85f 100644
--- a/test/runtests.jl
+++ b/test/runtests.jl
@@ -12,6 +12,7 @@ with_SCS(iters; accel=1, eps=1e-10) =
     acceleration_lookback=accel, eps=eps, warm_start=true)
 
 include("1703.09680.jl")
+include("actions.jl")
 include("1712.07167.jl")
 include("SOS_correctness.jl")
 include("1812.03456.jl")