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In the cohomology basis $h_1,\dots ,h_n$ supplied by the projective lines in the summands, the intersection form isFor each permutation $\sigma \in S_n$, the linear map $h_i\mapsto h_{\sigma (i)}$ preserves this pairing. Distinct permutations induce distinct maps, and their compositions agree with permutation composition. Hence the permutation matrices give a subgroup $G\cong S_n$ of the isometry group of $Q_X$.

To realize these isometries smoothly, describe $X$ as a punctured $S^4$ with $n$ punctured copies of ${\mathbb{CP}}^2$ attached to its boundary spheres. An orientation-preserving ambient isotopy of $S^4$ can interchange any two chosen disjoint attachment balls while carrying their parametrizations along with them: move their centers along disjoint arcs and extend the motion to small balls. There is room to make these arcs disjoint in dimension four. Extend the endpoint diffeomorphism across the two exchanged punctured copies using the chosen identifications of those identical oriented summands; use the identity on the remaining copies. Collar coordinates make the maps fit smoothly along the gluing spheres. This exchanges the two line classes without changing their signs. Since transpositions generate $S_n$, composing these diffeomorphisms realizes each element of $G$. Pullback permutes the cohomology basis by the inverse permutation, which still realizes the same subgroup and all its elements. These are permutation diffeomorphisms of identical connected summands.