  We prove that every homomorphism from the fundamental group of a planar Peano
continuum to the fundamental group of a planar or one-dimensional Peano
continuum is induced by a continuous map up to conjugation.
  This is then used to provide a family of uncountable many planar Peano
continua with pairwise non-isomorphic fundamental groups all of which are not
homotopy equivalent to a one-dimensional space.
