I was reading and doing some problems on wreath products. The groups are matrix group in my case.
Let $G_s=GL(2^s,R)$ denotes the group of invertible matrices over $R$ and suppose $M\le G_s$ for some positive number $s$ and some $R$ which is commutative with identity. Also suppose $C_2$ be the cyclic group of order $2$ generated by the permutation $(1,2)$. Then we consider the wreath product $M'=M\wr C_2 \le G_{s+1}$.
Now my question is
If $N$ and $N'$ are the normaliser of $M$ and $M'$ in $G_s$ and $G_{s+1}$ respectively, then is it true that $N'\le N\wr C_2$? Even if not true in general, is it true for some special case, say, when the base group of $M'$ is characteristic in $M'$?
Any help will be really appreciated. Thanks in advance.
Addendum: From the answer of Derek Holt, it is clear that the above is not true even if the base group of $M′$ is characteristic in $M'$.
For a counterexample, let $M = C_3 \times C_3 \le {\rm GL}(2,7)$, where $M$ consists of diagonal matrices, and $M' = M \wr C_2$.
Then $N'$ contains, for example, the permutation matrix that interchanges the first and third basis vectors and fixes the second and fourth, and this does not lie in ${\rm GL}(2,7) \wr C_2$.
Note that the base group of $M'$ has order $3^4$ and is of index $2$ in $M'$, so it is characteristic.