  \textit{Weak moonshine} for a finite group $G$ is the phenomenon where an
infinite dimensional graded $G$-module $$V_G=\bigoplus_{n\gg-\infty}V_G(n)$$
has the property that its trace functions, known as McKay-Thompson series, are
modular functions. Recent work by DeHority, Gonzalez, Vafa, and Van Peski
established that weak moonshine holds for every finite group. Since weak
moonshine only relies on character tables, which are not isomorphism class
invariants, non-isomorphic groups can have the same McKay-Thompson series. We
address this problem by extending weak moonshine to arbitrary width
$s\in\mathbb{Z}^+$. For each $1\leq r\leq s$ and each irreducible character
$\chi_i$, we employ Frobenius' $r$-character extension $\chi_i^{(r)} \colon
G^{(r)}\rightarrow\mathbb{C}$ to define \textit{width $r$ McKay-Thompson
series} for $V_G^{(r)}:=V_G\times\cdots\times V_G$ ($r$ copies) for each
$r$-tuple in $G^{(r)}:=G\times\cdots\times G$ ($r$ copies). These series are
modular functions which then reflect differences between $r$-character values.
Furthermore, we establish orthogonality relations for the Frobenius
$r$-characters, which dictate the compatibility of the extension of weak
moonshine for $V_G$ to width $s$ weak moonshine.
