  A variety of behaviors of entropy functions of random walks on finitely
generated groups is presented, showing that for any $\frac{1}{2}\leq
\alpha\leq\beta\leq1$, there is a group $\Gamma$ with measure $\mu$
equidistributed on a finite generating set such that \[\liminf\frac{\log
H_{\Gamma ,\mu}(n)}{\log n}=\alpha ,\qquad \limsup \frac{\log H_{\Gamma
,\mu}(n)}{\log n}=\beta .\] The groups involved are finitely generated
subgroups of the group of automorphisms of an extended rooted tree. The return
probability and the drift of a simple random walk $Y_n$ on such groups are also
evaluated, providing an example of group with return probability satisfying
\[\liminf\frac{{\log}|{\log P}(Y_n=_{\Gamma}1)|}{\log n}=\frac{1}{3},\qquad
\limsup\frac{{\log}|{\log P}(Y_n=_{\Gamma}1)|}{\log n}=1\] and drift satisfying
\[\liminf\frac{\log {\mathbb{E}}\|Y_n\|}{\log n}=\frac{1}{2},\qquad
\limsup\frac{\log {\mathbb{E}}\|Y_n\|}{\log n}=1.\]
