Let $R$ be a local (commutative, associative) ring over a field of zero characteristic.  (My typical examples are $k[[x_1,..,x_p]]/I$, $k\{x_1,.,x_p\}/I$, $C^\infty(\Bbb{R}^p,0)$.  If it helps one can assume $R$ to be Henselian.) 

I'd like to think of the ($k$-linear) ring automorphisms, $Aut(R)$, as the local changes of coordinates, "$Aut(Spec(R))$". The two objects certainly coincide  if $R$ is the localization of an affine ring.<br> 
More generally, let $S=k[x_1,..,x_p]/I$, let $S\subseteq R\subseteq \hat{S}$, the completion with respect to $(x_1,..,x_p)$. Then the two objects coincide for $R$.

But for $R=C^\infty(\Bbb{R}^p,0)$ there are endomorphisms which do not come from the local maps of coordinates. See <a href="https://www.ems-ph.org/journals/show_pdf.php?issn=0034-5318&vol=12&iss=1&rank=3">Page 5</a>. (In this particular one has endomorphism, not an automorphism. Still, it is not clear that here $Aut(R)=``Aut(Spec(R))"=Aut(\Bbb{R}^p,0)$, the later is the group of germs of local diffeomorphisms).

1. Suppose $R$ is "geometric enough", so that one can speak of $Spec(R)$, its local coordinates, a local change of them. Does every local change of coordinates (that preserves the origin) extend to an automorphism of $R$? 

2. For which "geometric" rings rings $Aut(R)=``Aut(Spec(R))"$? (i.e. the group of all the automorphisms of $R$ vs the group of the local coordinate changes in $Spec(R)$.) what is the official notation for the "geometric" subgroup $``Aut(Spec(R))"$ of $Aut(R)$?  (The notation $Aut(Spec(R))$ is somewhat heavy/lengthy.)

Any paper/review on the state of the art in this direction?