**Edit** I noticed that in Fiore-Leinster preliminate the condition (free monoidal category of an isomorphism $ \alpha: A \otimes A \to A $) is different from what is written in the preliminary question, so I reworked my answer substantially.

In a Monoidal category $\mathcal{C}$ consider (a non empty) class of sections of the type $\beta: A\to A\otimes A$ and let $\Sigma$ its tensor product closure (finite tensor products of some morphisms of type $ \beta $ of the choose class and some identities).

From the article "Note on monoidal localizations " by B. Day (link text) the category of fraction $\mathcal{C}_\Sigma$
is (naturally) a monoidal category.

let $P: \mathcal{C}\to \mathcal{C}_\Sigma$ the natural functor.

The elements of $\Sigma$ are all monomorphisms (are sections) , and if $\Sigma$ admits a calculus of left fractions the canonical functors $P$ is faithful (see "Categories" H Shubert, 12.9.6(a), p.261). THen $\mathcal{C}$-$Aut(X)$ is a subgroup of $\mathcal{C}_\Sigma$-$Aut(X)$ (because $P$ is faithful).

Now consider the Monoidal category $[A, \alpha, \beta]$ free on (the condition):

"one object $A$ and on two morphisms $\alpha: A\otimes A\to A$, $\beta: A\to A\otimes A$, with $\alpha\circ \beta=1_A$".

This category has the following universal property: for any monoidal categories $\mathcal{C}$ with choose morphisms $a: X\otimes X\to X,\ b: X\to X\otimes X$ with $a\circ b=1_X$ there exists a unique strict monoidal functor $F_{a,b}: [A, \alpha, \beta]\to \mathcal{C}$ with $F(\alpha)=a,\ F(\beta)=b$.

Now in $[A, \alpha, \beta]$ consider the tensor closure $\Sigma$ of the section $\beta$,

and let $P:[A, \alpha, \beta]\to [A, \alpha, \beta]_\Sigma$ the category of fractions.

the category $[A, \alpha, \beta]_\Sigma $ has the universal property of the monoidal category on one isomorphisms

$\beta: A\to A\otimes A$ as in the FIore-Leinster article, then
$F\cong [A, \alpha, \beta]_\Sigma$-$Aut(A)$.

Now, **IF** $\Sigma$ admit a calculus of left fraction then $P$ is faithful and
$[A, \alpha, \beta]$-$Aut(A)$ is isomorphic to a subgroup of $F$.

P.S. I seems that $\Sigma$ admit a calculus of left fraction, but I have not checked it in detail