(Probably a silly question, but..)

Consider the ring $R=k[[x_1,\dots,x_n]]/I$, (e.g. char(k)=0), and its subring, $R_1$, generated by some of $x_i$'s. In general, an automorphism of $R_1$ does not extend to that of $R$. (Even in the case of Noether normalization, right?) Does the extension hold for an automorphism that involves only "high order terms"? (i.e. an automorphism of $R_1$, whose N-jet is identity for N big enough.)

Even if this is not true in general, is it true for some nice rings? (pure dimensional or Cohen Macaulay)

What if the ring in question is just local but not necessarily complete?