Most of the nuclear fusion in the explosion of fusion bombs occurs inside the bomb within the hollow sphere (or a hollow cylindrical body) that constitutes the fission bomb. This is based on the fact that when the critical mass is exceeded, most of the inner energy in the core fission in the nuclear bomb, viewed segmentally, goes inwards and thus a higher pressure and a higher temperature build up inside, which are the conditions for nuclear fusion. The pressure and temperature on the outside of the bombs that we can produce are usually not sufficient for this.
At least that's the theory. This theory is taken up, for example, in the implosion design of fission bombs:
(The Original Image Source)
Note: the fusion core is usually a compressed enriched gas mixture of Deuterium $\operatorname{D} = \operatorname{_{1}^{2}H}$ and Tritium $\operatorname{T} = \operatorname{_{1}^{3}H}$ (Isotopes Of Hydrogen) and the fission explosive body is usually composed of uranium isotopes and or plutoniom isotopes.
Even if this is a common design, there are also variants in which an absolute maximum of nuclear fusion is attempted. In these experiments, attempts are also made to defend small parts of the shell during nuclear fusion. Attempts are also being made to use small parts of the hull in nuclear fusion.
To do this, you build the shells of the nuclear bomb out of lithium or lithium deuteride. Some of the Lithium Isotopes (Lithium-$6$ ($\operatorname{_{3}^{6}Li}$) and Lithium-$7$ ($\operatorname{_{3}^{7}Li}$)) can fuse with Deuterium $\operatorname{D}$ to form Helium-$4$ ($\operatorname{_{4}^{2}H}$) for this see see reaction Formulas $1$ and $2$. In addition, the Lithium-$6$ Deuteride $\operatorname{_{3}^{6}LiD}$ (This sometimes occurs, for example, in the Teller–Ulam-Design.) should also serve as a neutron source for the fission bomb (for this see Formula $3$).However, these Lithium Isotopes can also react directly with a neutron (for this see Formulas $4$ und $5$):
$$
\begin{align*}
\operatorname{_{3}^{6}Li} + \operatorname{D} = \operatorname{_{3}^{6}Li} + \operatorname{_{1}^{2}H} &\to 2 \cdot \operatorname{_{2}^{4}He} + 22.4 ~\mathrm{MeV} \tag{1}\\
\operatorname{_{3}^{7}Li} + \operatorname{D} = \operatorname{_{3}^{7}Li} + \operatorname{_{1}^{2}H} &\to 2 \cdot \operatorname{_{2}^{4}He} + n + 15.1 ~\mathrm{MeV} \tag{2}\\
\end{align*}
$$
$$
\begin{align*}
\operatorname{_{3}^{6}LiD} + n = \operatorname{_{3}^{6}Li_{1}^{2}H} + n &\to \operatorname{_{2}^{4}He} + \operatorname{_{1}^{3}H} + \operatorname{_{1}^{2}H} + 17.59 ~\mathrm{MeV} = \operatorname{_{2}^{4}He} + \operatorname{T} + \operatorname{D} + 17.59 ~\mathrm{MeV} \tag{3}\\
\end{align*}
$$
$$
\begin{align*}
\operatorname{_{3}^{6}Li} + n &\to \operatorname{_{2}^{4}He} + \operatorname{_{1}^{3}H} + 4.78 ~\mathrm{MeV} = \operatorname{_{2}^{4}He} + \operatorname{T} + 4.78 ~\mathrm{MeV} \tag{4}\\
\operatorname{_{3}^{7}Li} + n &\to \operatorname{_{2}^{4}He} + \operatorname{_{1}^{3}H} + n - 2,46 ~\mathrm{MeV} = \operatorname{_{2}^{4}He} + \operatorname{T} + n - 2,46 ~\mathrm{MeV} \tag{5}\\
\end{align*}
$$
All of this is still happening inside the nuclear bomb, but it's not necessarily inside the fusion device.