Primitive Wreath Product action I am a little confused about the primitive action of the wreath product as when to use the inverse and whether to use left or right action.
Let $H, K$ be groups and $K$ acts on $\Delta$, the wreath product of $H$ and $K$ is $H wr _{\Delta} K:= H^{\Delta} \rtimes K$.
There are two ways to interpret $H^{\Delta}$ as $\{(h_{\delta})_{\delta \in \Delta} \}$ or $Fun(\Delta ,H)$.
The action of $K$ on $H^{\Delta}$ is to permute the coordinates, so (first question:) could I use left action here and without inverse: $k \cdot (h_{\delta})_{\delta \in \Delta}=(h_{k\delta})_{\delta \in \Delta}$ or $(k \cdot f )(\delta)=f(k\delta)$, $f \in Fun(\Delta ,H)$.
Then we have $H$ acts on $\Gamma$, we can define the action of $H wr _{\Delta} K$ on $\Gamma ^{\Delta}$. Again (second question:) could I use left action here:  $(((h_{\delta})_{\delta \in \Delta} ,k ) \cdot \phi )(\delta ') =h_{k^{-1}\delta '} \phi (k^{-1}\delta ') $. ( I think inverse here are necessary as otherwise it will fail the associativity of the action.)
If the above the definitions are correct, when I check the association of the action, I have:
(i) $ (((g_{\delta})_{\delta \in \Delta} ,l)((h_{\delta})_{\delta \in \Delta} ,k ) \cdot \phi )(\delta )= ((g_{\delta})_{\delta \in \Delta} ,l) \cdot h_{k^{-1}\delta } \phi (k^{-1}\delta )=g_{k^{-1} l^{-1}\delta} h_{k^{-1}\delta}\phi( k^{-1} l^{-1} \delta)$
(ii) $(((g_{\delta} h_{l\delta})_{\delta \in \Delta}, lk ) \cdot \phi )(\delta ) = g_{k^{-1} l^{-1}\delta} h_{k^{-1}\delta}\phi( k^{-1} l^{-1} \delta)$
3rd: So the action of $k$ is to precompose $\delta$ with $k^{-1}$, but exactly is the action of $h$, how do we pick the subscripe $\delta$? i.e. I thought whenever $h$ acts on $\phi$, for each $\delta$, we get the corresponding $h_{\delta}$ and let it act on $\phi(\delta)$, then in (i) above, when $((g_{\delta})_{\delta \in \Delta} ,l)$ acts on $\phi (k^{-1}\delta )$, do we not have to regard $\phi (k^{-1}\delta )$ as a new map $\phi '(\delta )=\phi (k^{-1}\delta )$ and then it will be $g_{ l^{-1}\delta} \phi'( l^{-1} \delta)=g_{l^{-1}\delta} \phi( k^{-1} l^{-1} \delta)$ instead of $g_{k^{-1} l^{-1}\delta} \phi( k^{-1} l^{-1} \delta)$?
4th question: I think it would also work if we take right action and have inverse in both actions. Is there a standard way to specfify the action.
Sorry about such a long question. I would really appreciate if someone could clarify my confusion!
 A: I explain the wreath product with my notations and I hope it can be helpful.
Suppose that $H$ and $K$ be groups and $K$ acts on a non-empty set $\Delta$. The action of $K$ on $Fun(\Delta ,H)$ can be defined as follows 
$$\forall k\in K \  \forall f\in Fun(\Delta ,H) \ f^k(\delta):=f(\delta^{k^{-1}}).$$
Also, the action of $K$ on $H^{\Delta}$, where $H^{\Delta}$ is $\{(h_{\delta})_{\delta \in \Delta} \}$, can be defined as
$$\forall k\in K \  \forall (h_{\delta})_{\delta \in \Delta}\in H^{\Delta} \ ((h_{\delta})_{\delta \in \Delta})^k:=(h_{\delta^k})_{\delta \in \Delta}.$$
The wreath product of $H$ and $K$, denoted by $G$, is $H \ wr _{\Delta} K:= Fun(\Delta ,H)\rtimes K$ or $H \ wr _{\Delta} K:=H^{\Delta} \rtimes K$.
Now, suppose that $H$ acts on the set $\Gamma$. The product action of $G$ on $Fun(\Delta ,\Gamma)$ (or equivalently on $\Gamma^\Delta$) can be defined as follows 
$$\forall (f,k)\in G=Fun(\Delta ,H)\rtimes K  \quad \forall\phi\in Fun(\Delta ,\Gamma) \quad \phi^{(f,k)}(\delta):=\phi(\delta^{k^{-1}})^{f(\delta^{k^{-1}})}$$
or equivalently as
$$\forall ((h_{\delta})_{\delta \in \Delta} ,k )\in G=H^{\Delta} \rtimes K \quad  \forall (\gamma_{\delta})_{\delta \in \Delta}\in \Gamma^\Delta \quad ((\gamma_{\delta})_{\delta \in \Delta})^{((h_{\delta})_{\delta \in \Delta} ,k )}:=((\gamma_{\delta^k})^{h_{\delta^k}})_{\delta \in \Delta}.$$
It is easy to see that the above actions are associative.
In your questions, the definition $k \cdot (h_{\delta})_{\delta \in \Delta}:=(h_{k\delta})_{\delta \in \Delta}$ is true, but you should define $(k \cdot f )(\delta):=f(k^{-1}\delta)$ because of the associativity of the action.
In the above definitions, you can replace the powers by left (or right) actions with possibly small changes.
