= Solution
The <inertia tensor> is $I_{ij}=\int(r^2\delta_{ij}-x_ix_j)\,dm$. The mixed moments vanish by reflection in $x$ or $y$. Every diagonal integrand is even in $z$, so its <integral> over the upper hemisphere is half its <integral> over the corresponding full ball of mass $2M$. A uniform ball's coordinate moments are each $(2M)a^2/5$, obtained by spherical symmetry and $\langle r^2\rangle=3a^2/5$. Hence the hemisphere has each coordinate <second moment> $Ma^2/5$ and
$$
\boxed{I_O=\frac25Ma^2I.}
$$
The tensor <parallel axis theorem> gives $I_O=I_C+M(|d|^2I-dd^t)$ with $d=(0,0,3a/8)$. Therefore
$$
\boxed{I_C=Ma^2\operatorname{diag}\!\left(\frac{83}{320},\frac{83}{320},\frac25\right).}
$$
Translation along the symmetry axis changes the two transverse moments but leaves the axial one unchanged.
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