  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.
