= Solution
An <orientation of a vector bundle> of real rank $d$ over $R$ is a coherent choice of generator of $H^d(E_b,E_b\setminus\{0\};R)$ in every fibre. For a complex bundle of rank $m$, a complex basis gives the real basis $(v_1,iv_1,\ldots,v_m,iv_m)$. A complex change of basis has positive real determinant $|\det_{\mathbb C}A|^2$, so these local choices agree and give the <canonical orientation of a complex vector bundle> over $\mathbb Z$, hence over every commutative ring $R$.
Let $D(E)$ and $S(E)$ be the disk and sphere bundles of an $R$-oriented rank-$d$ bundle over compact $B$. A <Thom class> $u_E\in H^d(D(E),S(E);R)$ restricts to the chosen generator on every fibre. The <Thom isomorphism theorem> states that
$$
H^q(B;R)\xrightarrow{\sim}H^{q+d}(D(E),S(E);R),
\qquad a\longmapsto\pi^*a\smile u_E.
$$
The <Euler class of a vector bundle> is $e(E)=s^*u_E\in H^d(B;R)$ for the zero section $s$. Substituting the Thom isomorphism into the long exact sequence of the pair $(D(E),S(E))$ gives the <Gysin sequence of a sphere bundle>
$$
\cdots\to H^{q-d}(B;R)\xrightarrow{\smile e(E)}H^q(B;R)\to H^q(S(E);R)\to H^{q-d+1}(B;R)\to\cdots.
$$
Apply this to the <Hopf fibration> and induct on $n$. If $x=c_1(\mathcal O(1))$ has degree two, the result is the <cohomology ring of complex projective space>
$$
H^*(\mathbb{CP}^n;\mathbb Z)\cong\mathbb Z[x]/(x^{n+1}).
$$
The <Künneth theorem> over a principal ideal domain gives a natural short exact sequence with tensor-product and Tor terms; the sequence splits, though not naturally. Here all groups are free, so the Tor term vanishes and
$$
H^*(\mathbb{CP}^n\times\mathbb{CP}^n;\mathbb Z)
\cong\mathbb Z[x,y]/(x^{n+1},y^{n+1}).
$$
If a graded-ring automorphism sends $x$ to $ax+by$, then $(ax+by)^{n+1}=0$. The coefficients of the nonzero mixed monomials force $ab=0$; the same applies to the image of $y$. Invertibility then forces a signed permutation matrix on the basis $x,y$. Conversely, complex conjugation on either factor changes the sign of its degree-two generator, and swapping the factors exchanges $x$ and $y$. Thus the realizable group is
$$
\boxed{(\mathbb Z/2)^2\rtimes S_2,}
$$
the group of all signed $2$ by $2$ permutation matrices, as described by the <cohomology automorphisms of a product of two complex projective spaces>.
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