We consider a mechanism design problem for the provision of a binary excludable public good. We characterize the augmented serial rules (Ohseto, Econ Theory 26:589-606 2005) by strategy-proofness, envy-freeness, and access independence. This result is the positive answer to the first open question posed by Ohseto (Econ Theory 26:589-606, 2005). We also show that, in addition to the augmented serial rules, there exists a rule satisfying strategy-proofness, symmetry, access independence, and non-bossiness. This result is the negative answer to the second open question posed by Ohseto (Econ Theory 26:589-606, 2005).