  In this article, we establish an $L^2$ extension theorem for Nakano
semi-positive singular Hermitian metrics on holomorphic vector bundles, and the
strong openness and stability properties of the multiplier submodule sheaves
associated to Nakano semi-positive singular Hermitian metrics on holomorphic
vector bundles.
  We solve affirmatively a question of Lempert on the preservation of Nakano
semi-positivity under limit of an increasing metrics based on
Deng-Ning-Wang-Zhou's characterization of Nakano positivity.
