  Let $f_{0,\infty}=\{f_n\}_{n=0}^{\infty}$ be a sequence of continuous
self-maps on a compact metric space $X$. Firstly, we obtain the relations
between topological sequence entropy of a nonautonomous dynamical system
$(X,f_{0,\infty})$ and that of its finite-to-one extension. We then prove that
the topological sequence entropy of $(X,f_{0,\infty})$ is no less than its
corresponding measure sequence entropy if $X$ has finite covering dimension.
Secondly, we study the supremum topological sequence entropy of
$(X,f_{0,\infty})$, and confirm that it equals to that of its $n$-th
compositions system if $f_{0,\infty}$ is equi-continuous; and we prove the
supremum topological sequence entropy of $(X,f_{i,\infty})$ is no larger than
that of $(X,f_{j,\infty})$ if $i\leq j$, and they are equal if $f_{0,\infty}$
is equi-continuous and surjective. Thirdly, we investigate the topological
sequence entropy relations between $(X,f_{0,\infty})$ and
$(\mathcal{M}(X),\hat{f}_{0,\infty})$ induced on the space $\mathcal{M}(X)$ of
all Borel probability measures, and obtain that given any sequence, the
topological sequence entropy of $(X,f_{0,\infty})$ is zero if and only if that
of $(\mathcal{M}(X),\hat{f}_{0,\infty})$ is zero; the topological sequence
entropy of $(X,f_{0,\infty})$ is positive if and only if that of
$(\mathcal{M}(X),\hat{f}_{0,\infty})$ is infinite. By applying this result, we
obtain some big differences between entropies of nonautonomous dynamical
systems and that of autonomous dynamical systems. Finally, we study whether
multi-sensitivity of $(X,f_{0,\infty})$ imply positive or infinite topological
sequence entropy.
