Given $X$ a topological space, we consider $\mathcal{F}$ the class of all continuous maps $f:X\to H$ where $H$ is a topological group... (edited) and $|H|\le |X|$
If $f,g\in\mathcal{F}$, say $f:X\to H_f$ and $g:X\to H_g$, consider the relation defined by $f\sim g$ if there exists a homeomorphism $\psi : H_f\to H_g$ such that $g = \psi\circ f$.
I want to show that the quotient $\mathcal{F}/\sim$ is a set which cardinality is not greater than $2^{2^{|X|}}$
Any hint?
I shall consider only the case when $X$ is infinite. It seems the following.
Since each topology on the set $H$ is a subset of $2^H$, we see that there are at most $\le 2^{2^|H|}$ group topologies on the set $H$. Since a multiplication on a set $H$ is a map from $H^{H\times H}$, we see that there are at most $|X|^{|X|\times |X|}\cdot |X|\cdot 2^{2^{|X|}}=2^{2^{|X|}}$ non-isomorphic topological groups of size at most $|X|$. Then $|\mathcal F/\sim|\le |(2^{2^{|X|}})^{|X|}|=2^{2^{|X|}}$.
PS. It is more natural to define such equivalence relations by topological isomorphisms instead of homeomorphisms.