Induced mappings between quotient spaces of symmetric products of continua

Given a continuum X   and n∈Nn∈N. Let H(X)∈{2X,C(X),Fn(X)}H(X)∈{2X,C(X),Fn(X)} be a hyperspace of X  , where 2X2X, C(X)C(X) and Fn(X)Fn(X) are the hyperspaces of all nonempty closed subsets of X, all subcontinua of X and all nonempty subsets of X with at most n   points, respectively, with the Hausdorff metric. For a mapping f:X→Yf:X→Y between continua, let H(f):H(X)→H(Y)H(f):H(X)→H(Y) be the induced mapping by f  , given by H(f)(A)=f(A)H(f)(A)=f(A). On the other hand, for 1⩽m