In Geometric Topology - Localization, Periodicity, and Galois Symmetry by Dennis Sullivan, we can read that there is a decomposition $$B\mathrm{SL}(\mathbb S_{(p)})\times K((\mathbf Z_{(p)})^\times)\xrightarrow{\simeq}B\mathrm{GL}(\mathbb S_{(p)})$$ induced by fiberwise join of associated bundles. I can guess that this decomposition is not $E_\infty$, for example for $p=2$, the first Stieffel Whitney class coming from $BO$ does not vanish under the application of Dyer Lashof operations.

We can find useful information about the action of the Dyer Lashof algebra by using the fact that $B\mathrm{SL}(\mathbb S_{(p)})\to B\mathrm{GL}(\mathbb S_{(p)})$ is an $E_\infty$-map and the (highy nontrivial) fact that $B\mathrm{SL}(\mathbb S_{(p)})$ is the $p$-localisation of $B\mathrm{SL}(\mathbb S)=BSF$, the homology of which is very well-known.

Is there any reference on the Dyer Lashof action on $H_1(B\mathrm{GL}(\mathbb S_{(p)}))=(\mathbf Z_{(p)})^\times$ ? Note that I am actually mostly interested in the case of $p$-completed spherical bundles, where all of the above discussion works but with $(p)$ replaced by $p$.

Probably, the case $p=2$ might be a bit different from the others... I could not work this calculation out myself.