Let \(R\subseteq S\) be an extension of commutative rings with 1 and denote by \([R,S]\) the set of \(R\)-subalgebras of \(S.\) We say the extension \(R\subseteq S\) has the property FIP if \([R,S]\) is finite and that it has FCP if each chain(i.e. a set of elements that are pairwise comparable with respect to inclusion) of \([R,S]\) is finite. The main aim of the paper is to study the transfer of the FIP and FCP properties between an extension \(R\subseteq S\) and the induced extension between the Nagata rings, \(R(X)\subseteq S(X).\) It is shown that if \(R(X)\subseteq S(X)\) has FIP( resp. FCP) then \(R\subseteq S\) has FIP(resp. FCP). The main result is Theorem 3.9 which shows that \(R\subseteq S\) has FCP iff \(R(X)\subseteq S(X)\) has FCP. On the other hand, since in general if \(R\subseteq S\) has FIP it doesn’t follow that \(R(X)\subseteq S(X)\) has FIP, several cases in which this is true are investigated. For example it is shown that if \(R\subseteq S\) is a seminormal integral extension, it has FIP iff \(R(X)\subseteq S(X)\) has FIP.