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Roberts and Schmidt have developed a theory of new vectors for generic irreducible smooth representations of $\operatorname{PGSp}_4(F)$ for $F$ a nonarchimedean local field, using the "paramodular subgroups".

Has this theory been generalized to representations of $\operatorname{GSp}_4(F)$ that have nontrivial central character (and thus don't factor through $\operatorname{PGSp}_4(F)$)?

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I'm adding an answer to this very old question of mine, since someone just contacted me about it. The problem is rather comprehensively solved in this 2019 preprint of Taeko Okazaki:

Takeo Okazaki, Local Whittaker-Newforms for GSp(4) matching to Langlands parameters, https://arxiv.org/abs/1902.07801

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