= Solution
For $X\in\mathfrak h$, the <fundamental vector field> of the restricted <Lie group action> is $(iX)_M$. If $\phi$ is the original <moment map>, then
$$
d\langle\pi\circ\phi,X\rangle_H
=d\langle\phi,iX\rangle_G
=\iota_{(iX)_M}\omega
=\iota_{X_M}\omega.
$$
This is the defining <Hamiltonian action> identity, in the convention $\iota_{X_H}\omega=dH$.
If equivariance is included in the definition of a <moment map>, it is also preserved: the <dual map> $\pi=i^*$ intertwines the <coadjoint actions> of $H$, since $i(\operatorname{Ad}_hX)=\operatorname{Ad}_h(iX)$. Thus
$$
\boxed{\phi_H=i^*\phi=\pi\circ\phi}
$$
is a <moment map> for the restricted <Hamiltonian action> of the <Lie subgroup> $H$.
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