= Solution
At a current value $\beta$, put
$$
\eta_i=x_i^T\beta=g(\mu_i),
\qquad
W_{ii}=
\frac1{a_i\sigma^2V(\mu_i)\{g'(\mu_i)\}^2},
$$
and define the working response
$$
z_i=\eta_i+(Y_i-\mu_i)g'(\mu_i).
$$
Then the displayed formulas in the question can be written compactly as
$$
\mathcal I(\beta)=X^TWX,
$$
and
$$
U(\beta)=X^TW(z-X\beta).
$$
Indeed, the $j$th component of the latter is
$$
\sum_{i=1}^n
\frac{X_{ij}(Y_i-\mu_i)}
{a_i\sigma^2V(\mu_i)g'(\mu_i)},
$$
which is precisely the stated score.
Substitution into the Fisher-scoring update gives
$$
\beta^{\rm new}
=\beta+(X^TWX)^{-1}X^TW(z-X\beta)
=(X^TWX)^{-1}X^TWz.
$$
By part (b), this is the <weighted least squares> fit of the working response $z$ on $X$. Both $z$ and $W$ are recomputed from the current fitted means after every update. Fisher scoring for a <generalized linear model> is therefore the <iteratively reweighted least squares> algorithm.
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