According to Theorem 9, polynomials
invertible in $k$-radicals for $8\leq k\leq 14$ are compositions of
power polynomials, Chebyshev polynomials, polynomials of degree at
most $k$, polynomials of degree 10 with monodromy group isomorphic
to $P\Gamma L_{2}(9)$ described in Sect. 6.2 and
polynomials of degree 15 with monodromy group isomorphic to
$PGL_{4}(2)$ with its natural action either on the 15 points or on the
15 hyperplanes of the thee-dimensional projective space over the
field $F_{2}$.

Polynomials of degree 15 with monodromy group isomorphic to
$PGL_{4}(2)$ can have one of the following passports
([Jones and Zvonkin2002]; [Adrianov1997]):
$[2^{6}1^{3},2^{4}1^{7},2^{4}1^{7}]$,
$[4^{3}2^{1}1^{1},2^{4}1^{7}]$, $[4^{2}2^{2}1^{3},2^{6}1^{3}]$, $[6^{1}3^{2}2^{1}1^{1},2^{4}1^{7}]$.

Such polynomials had been investigated in
[Cassou-Noguès and Couveignes1999] in context of finding pairs of
polynomials $g,h$ such that the curve $g(x)=h(y)$ is reducible. In
[Cassou-Noguès and Couveignes1999] it is proved that a polynomial
with monodromy group isomorphic to $PGL_{4}(2)$ and passport
$[2^{6}1^{3},2^{4}1^{7},2^{4}1^{7}]$ can be brought by an affine change of
variables to the form

where $a$ is one of the two roots of the equation $a^{2}-a+4=0$ and
$t$ is a complex number.

This result is mentioned there only briefly and leaves several
questions unanswered:

Do all the polynomials from the families $g_{t}^{a}$ have monodromy
group $PGL_{4}(2)$ with action on the points or on the hyperplanes of
the space $P^{3}(F_{2})$ depending on the choice of $a$ for all
parameters $t\neq 0$?

Can all the polynomials of degree 15 with monodromy group isomorphic
to $PGL_{4}(2)$ can be brought by an affine change of variables to the
form $g_{t}^{a}(x)$ for some $t$ and some choice of $a$? In particular
do all the polynomials with monodromy group $PGL_{4}(2)$ with
passports $[4^{3}2^{1}1^{1},2^{4}1^{7}]$, $[4^{2}2^{2}1^{3},2^{6}1^{3}]$,
$[6^{1}3^{2}2^{1}1^{1},2^{4}1^{7}]$ correspond to some values of the parameter?
It is certainly true for some of them. For instance for $t=75/4$ the
polynomial $g_{t}^{a}$ has passport $[4^{2}2^{2}1^{3},2^{6}1^{3}]$ and dessin
d’enfant

(or its reflection for the other choice of $a$).

For $t=-5/4$ the polynomial has the passport $[6^{1}3^{2}2^{1}1^{1},2^{4}1^{7}]$ and dessin d’enfant

(or its reflection for the other choice of $a$).

For $t=-405/4$ it has the passport $[4^{3}2^{1}1^{1},2^{4}1^{7}],$ and
dessin d’enfant

(or its reflection for the other choice of $a$).

We thank Michael Zieve, who communicated these special values of $t$
to us in personal correspondence.