Let $\mathcal A$ and $\mathcal B$ be unital Banach algebras and $T:\mathcal A\longrightarrow B$ be a unital continuous homomorphism. Put $\mathcal J=\operatorname{Ker}T$. Let $\operatorname{Fred}_T(\mathcal A)=\{x\in\mathcal A|T(x)$ is invertible in $\mathcal B\}$ and $\operatorname{Fred}^0_T(\mathcal A)=\{x+|x$ is invertible in $\mathcal A,k\in J\}$. In this note, we prove that if $T$ has Property $(F)$, then $\operatorname{Fred}_T(\mathcal A)\cap GL(\mathcal A)=\operatorname{Fred}^0_T(\mathcal A)$ iff ltsr $(\mathcal J)=1$.