How to solve Transfer function with step function appplied to the input?
The Laplace transform \$F(s)\$ of the response of the system to a unit step input is   $$ F(s) = \frac{880. (1. s+6.81818)}{s (s+6.) (s+10.)} $$

The Laplace transform \$E(s)\$ of the input signal (unit step) is 

$$ E(s) = \frac{1}{s} $$

The transfer function \$G(s)\$ relates the input and output by \$F(s)=G(s) E(s)\$.

Therefore, $$G(s)=\frac{F(s)}{E(s)} = \frac{880. (1. s+6.81818)}{(s+6.) (s+10.)}$$