$\sigma$-weak amenability of Banach algebras

Let $\mathcal{A}$ be a Banach algebra, $\sigma$ be continuous homomorphism on $\mathcal{A}$ with $\overline{\sigma(\mathcal{A})}=\mathcal{A}$. The bounded linear map $D : \mathcal{A}\to\mathcal{A}^*$ is $\sigma$-derivation, if
$$D(ab) = D(a) \sigma(b) + \sigma(a)  D(b)\quad (a, b\in \mathcal{A}).$$
We say that A is $\sigma$-weakly amenable, when for each bounded derivation $D : \mathcal{A}\to\mathcal{A}^*$, there exists $a^*\in \mathcal{A}^*$ such that $D(a) = \sigma(a) a^*-a^*\sigma(a)$. For a commutative Banach algebra $\mathcal{A}$, we show $ \mathcal{A}$ is $\sigma$-weakly amenable if and only if every $\sigma$-derivation from $\mathcal{A}$ into a $\sigma$-symmetric Banach $ \mathcal{A}$-bimodule $X$ is zero. Also, we show that a commutative Banach algebra $ \mathcal{A}$ is $\sigma$-weakly amenable if and only if $A^\#$ is $\sigma^\#$-weakly amenable, where $\sigma^\#(a + \alpha) = \sigma(a) +\alpha$.