Admissible morphisms for Shimura varieties with parahoric levels

In \textit{Shimuravariet\"{a}ten und Gerben} \cite{LR87}, Langlands and Rapoport developed the theory of pseudo-motivic Galois gerb and admissible morphisms between Galois gerbs, with a view to formulating a conjectural description of the $\bar{\mathbb{F}}_p$-point set of the good reduction of a Shimura variety with hyperspecial level, as well as to providing potential tools for its resolution. Here, we generalize, and also improve to some extent, their works to parahoric levels when the group is quasi-split at $p$. In particular, we show that every admissible morphism is conjugate to a special admissible morphism, and, when the level is special maximal parahoric, that any Kottwitz triple with trivial Kottwitz invariant, if it satisfies some obvious necessary conditions implied by the conjecture, comes from an admissible pair. As applications, we give natural effectivity criteria for admissible pairs and Kottwitz triples, and establish non-emptiness of Newton strata in the relevant cases. Along the way, we fill some gaps in the original work.