  Let $K$ be a field and $G$ be a group of its automorphisms endowed with the
compact-open topology. There are many situations, where it is natural to study
the category $Sm_K(G)$ of smooth (i.e. with open stabilizers) $K$-semilinear
representations of $G$. The category $Sm_K(G)$ is semisimple (in which case $K$
is a generator of $Sm_K(G)$) if and only if $G$ is precompact. In this note we
study the case of the non-precompact group $G$ of all permutations of an
infinite set $S$. It is shown that the categories $Sm_K(G)$ are locally
noetherian; the morphisms are `locally split'. Given a field $F$ and a subfield
$k\neq F$ algebraically closed in $F$, one of principal results describes the
Gabriel spectra (and related objects) of the categories $Sm_K(G)$ for some of
$G$-invariant subfields $K$ of the fraction field $F_{k,S}$ of the tensor
product over $k$ of the labeled by $S$ copies of $F$. In particular, the object
$F_{k,S}$ turns out to be an injective cogenerator of the category $Sm_K(G)$
for any $G$-invariant subfield $K$ of $F_{k,S}$. As an application, when
transcendence degree of $F|k$ is 1, a correspondence between the $G$-invariant
subfields of $F_{k,S}$ algebraically closed in $F_{k,S}$ and certain systems of
isogenies of `generically $F$-pointed' torsors over absolutely irreducible
one-dimensional algebraic $k$-groups is constructed, so far only in
characteristic 0. The only irreducible finite-dimensional smooth representation
of $G$ is trivial. However, there is an invariant `cross-ratio' subfield $K$ of
$k(S)$ such that the irreducible finite-dimensional smooth $K$-semilinear
representations of $G$ correspond to the irreducible algebraic representations
of $PGL_{2,k}$. In general, any smooth $G$-field $K$ admits a smooth $G$-field
extension $L|K$ such that $L$ is a cogenerator of $Sm_L(G)$.
