Automorphisms of Cuntz algebra

Suppose, $O_{\infty}$ is the cuntz algebra generated by the orthogonal isometries $\{S_i\}_{i\in \mathbb{N}}$,i.e. $S_i^*S_j=\delta_{ij}$ and $O_{\infty}=C^*(\{S_i\}_{i\in \mathbb{N}})$.

Then the span of $\{S_iS_j\}_{i,j\in \mathbb{N}}$ forms a Hilbert space $H$ with the inner product being $\langle u,v\rangle 1=v^*u$.

Given a unitary operator $U:H\to H$ so that $U$ commutes with the flip operator $\sigma: H\to H$ (defined by $\sigma(S_iS_j)=S_jS_i \forall i,j$),i.e. $U\sigma=\sigma U$, does there exist an automorphism $\phi$ of $O_{\infty}$ so that $\phi(x)=U(x) \forall x\in H$.

Note that, if we consider the Hilbert space $H'$ to be the span of $\{S_i\}_{i\in \mathbb{N}}$ with inner product defined similar as before, then $H=H'\otimes H'$ and if we take $U$ to be $V\otimes V$ where $V$ is a unitary on $H'$, then clearly the above statement is true for $U$.

I am trying to see if its true when we take the strong closure of the span of elements of the form $V\otimes V$($V$ is in $U(H')$) in $B(H)$ which is basically the commutant of $\sigma$.

It seemed to me that this is not always possible. Actually, I couldn't find $\phi$ when $U=\sigma$. So in that case, is there any nice characterization of the unitaries on $H$ for which this is possible?

Also, can anyone give a reference to any paper where automorphisms of $O_{\infty}$ has been studied? I only found papers on automorphisms of $O_n$ but not of $O_{\infty}$.