Let us fix a connected split reductive group $G$ over a local non-archimedean field $K$ with maximal torus $T$, then most notes including that of [Gross] defines the Satake isomorphism as $$\mathcal{S}:\mathcal{H}_T\otimes \mathbb{Z}[q^{-1/2},q^{1/2}]\xrightarrow{\cong}R(\hat{G})\otimes \mathbb{Z}[q^{-1/2},q^{1/2}],$$ where $q$ is the cardinality of the residue field of $K$, $\mathcal{H}_T$ is the Hecke algebra (of the torus) of functions which are bi-invariant under the action of $T(\mathcal{O}_K)$ and $R(\hat{G})$ is the Grothendieck ring of the category of complex representaions of the connected dual group $\hat{G}= {}^LG^\circ$. 

A nice consequence of this isomorphism discussed in the Propositon 6.4 of [Gross], is that the Satake isomorphism gives us a bijection between unramified representations of $G(K)$ and semisimple conjugacy classes in $\hat{G}(\mathbb{C})$.

Now I have also heard people discuss the Satake isomorphism for quasi-split reductive groups as giving a bijection between unramified representations of $G(K)$, and conjugacy classes of the Langlands dual group ${}^LG$ whose projection to $\hat{G}$ is semisimple and that to the Galois group is Frob. I had a few questions:

 1. Is the reformulation correct?

Assuming it is correct:

 2. What does projection to $\hat{G}$ mean when the Galois group is not normal? If I just take the set theoretic projection, couldn't a conjugacy class in ${}^LG$ project to different elements in $\hat{G}$ - what if some of these are semisimple and some are not?

3. If these two formulations are the same, how is Frob hiding in the formulation given in [Gross].

My best guess is that if we take conjugates $(g,\phi)$ and $(h,\psi)$ in ${}^LG=\hat{G}\rtimes \operatorname{Gal}(\bar{K},K)$ such that $\phi$ and hence also $\psi$ is Frob (modulo inertia), then $g$ and $h$ will either both be semisimple or both not (and vice versa) - or better $g$ and $h$ are also conjugate. But, just writing these down doesn't make it obvious.

Sorry if this is too much in one question, but thanks in advance for any help.  

References:

<cite authors="Gross, Benedict H.">_Gross, Benedict H._, On the Satake isomorphism, Scholl, A. J. (ed.) et al., Galois representations in arithmetic algebraic geometry. Proceedings of the symposium, Durham, UK, July 9-18, 1996. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 254, 223-237 (1998). [ZBL0996.11038](https://zbmath.org/?q=an:0996.11038).</cite>