MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).

Uniqueness is hopeless; let $k$ be any reduced ring and $R=k[x]$. Then $k\subset R$ and $R\subset R$ have the same group of units.
Existence is also hopeless in general: Let $S=Z/5Z$, and let $H$ be the subgroup conisting of 1 and 4.
Uniqueness is hopeless; let $k$ be any ring and $R=k[x]$. Then $k\subset R$ and $R\subset R$ have the same group of units.
Existence is also hopeless in general: Let $S=Z/5Z$, and let $H$ be the subgroup conisting of 1 and 4.