<p>We study a class of bilevel variational inequality problems over the solution sets of split variational inequality problems with multiple output sets in real Hilbert spaces, where the cost operators are quasimonotone. To solve this class of problems, we propose a strongly convergent algorithm that combines alternated inertial extrapolation with a self-adaptive step-size strategy. The proposed method guarantees strong convergence without requiring line search procedures or prior knowledge of problem-dependent parameters such as Lipschitz constants or strong monotonicity coefficients of the upper-level cost operator. Numerical experiments confirm the robustness and computational efficiency of the algorithm.</p>