It is well-known that in a semi-direct product of two groups $G = N\rtimes_\phi H$ the exponent of $G$ might be bigger than the lcm of the exponents of $N$ and $H$.
It sometimes does not, I give an example below. Do you know any condition that can be asked about $N$ and $\phi: H\to Aut(G)$ that ensures that the exponent of $G$ is the same as the exponent of $N$?
Even in the case where $H$ is a subgroup of inner automorphisms of $N$?
An example of this is when $N$ is nilpotent of class $c<p$ where $p$ is the exponent of $N$, and $H$ is a subgroup of $Aut(G)$ of inner automorphisms. Then $G = N\rtimes H$ is again nilpotent of class $c<p$ by a Theorem of Kaloujnine. As such it is a regular $p$-group and hence is of exponent $p$ again.