I am little confused about the definition of wreath product in section 4.1 of The Representation Theory of the Symmetric Group by Gordon James and Adalbert Kerber.
The definition is given as below.
Let $G$ be a group and $H$ a subgroup of $S_n$. Denoting by $G^{\mathbf{n}}$ the set of all mappings from $\mathbf{n} = \{1, \ldots, n\}$ into $G$:
$$ G^{\mathbf{n}} := \{f | f : \mathbf{n} \to G\}, $$
we put
$$ G \text{wr} H := G^{\mathbf{n}} \times H = \{(f; \pi) | f : \mathbf{n} \to G \text{ and } \pi \in H\} $$
Then the authors add as follows.
For $f \in G^{\mathbf{n}}$ and $\pi \in H$ we define $f_\pi := f \circ \pi^{-1}$
It is easy to check that for $\pi' \in H$
$$ (f_\pi)_{\pi'} = f_{\pi' \pi}. $$
Furthermore, we introduce a multiplication on $G^{\mathbf{n}}$ by
$$ (ff')(i) := f(i) \cdot f'(i), i \in \mathbf{n}. $$
Using this, one easily verifies that $G \text{ wr } H$ together with the composition defined by
$$ (f; \pi) (f'; \pi') := (ff^{'}_\pi; \pi \pi') $$
forms a group, the wreath product of $G$ by $H$.
In the definition given in the Wikipedia, $G^\mathbf{n}$ is defined as a direct product of $G$ with itself of order $n$ and the product with $H$ is a semidirect product.
Why are these two definitions different? Thanks.
PS. I have added more details at Eric's advice.
The definitions are exactly the same. By definition, if $A$ and $B$ are groups and $\varphi:B\to\operatorname{Aut}(A)$ is a homomorphism, their semidirect product is defined as the set $A\times B$ with multiplication defined by $$(a,b)\cdot(a',b')=(a\varphi(b)(a'),bb').$$ This is exactly the definition you quote, in the case $A=G^\mathbf{n}$, $B=H$, and $\varphi$ is defined by $\varphi(\pi)(f)=f_\pi$.