  The question of whether two words in a free group that induce the same
measure on every finite group as word maps are automorphic remains open. In
this work, we study words whose images as word maps on every finite group are
identical. We establish that two words in $F_2$, where one of the words is
$x^n$ or $[x,y]^n$ for $n \in \Z$, having same image on every finite group, are
endomorphic to each other. Furthermore, we demonstrate that if the word map
corresponding to a word $w_1 \in F_n$ has the same image as a test word $w_2
\in F_n$ on every finite group, then this is sufficient to ensure that $w_1$
and $w_2$ induce the same measure on every finite group.
