  For large $n$ we determine the maximum number of induced 6-cycles which can
be contained in a planar graph on $n$ vertices, and we classify the graphs
which achieve this maximum. In particular we show that the maximum is achieved
by the graph obtained by blowing up three pairwise non-adjacent vertices in a
6-cycle to sets of as even size as possible, and that every extremal example
closely resembles this graph. This extends previous work by the author which
solves the problem for 4-cycles and 5-cycles. The 5-cycle problem was also
solved independently by Ghosh, Gy\H{o}ri, Janzer, Paulos, Salia, and Zamora.
