A pentagon system of order n is a partition of the edges of \(K_ n\) into 5-cycles: and exists if and only if \(n\equiv 1, 5\) (mod 10) [A. Rosa, Casopis Pest. Mat. 91, 53-63 (1966; Zbl 0151.335)]. A pentagon system of order n is nested if there is a mapping \(\rho\) from \({\mathbb{P}}\) (the set of 5-cycles in the system) to the vertices of \(K_ n\) so that \(\{\rho(P)a, \rho(P)b, \rho(P)c, \rho(P)d, \rho(P)e:\;P=(a,b,c,d,e)\in {\mathbb{P}}\}=E(K_ n).\) For such a nested system it is necessary that \(n\equiv 1(mod 10)\) and it is shown here that if \(n\not\in \{111, 201, 221, 231, 261, 301, 381, 511, 581, 591, 621\},\) then the system exists. In the cases when \(n\equiv 5 (mod 10)\) such systems are impossible but it is shown that if two parallel classes of 5-cycles are deleted from \(K_ n\), then, provided \(n\not\in \{15, 25\},\) a nested pentagon system on the remaining graph exists. It is also shown that a nested pentagon system on the graph \(2K_ n\), \(n\geq 6\), exists if and only if \(n\equiv 0,1 (mod 5),\quad n\not\in \{10, 15, 30\}.\) Finally, the authors consider the existence of nested pentagon systems on \(2K_ n\), \(n\equiv 1 (mod 5),\) which are almost resolvable; that is, in which the 5-cycles can be partitioned into partial parallel classes each consisting of (n-1)/5 5- cycles. These are shown to exist for all admissible n except \(n\in \{26, 201, 221, 231, 261, 581, 591, 621\}\).