Could anybody point to me a good reference for defining the $\Gamma$-category associated to a permutative category (better also with some illustrative examples)?


Dan Freed in his [notes](https://www.ma.utexas.edu/users/dafr/M392C-2012/Notes/lecture19.pdf) does provide a definition but many details are left out.



The $\Gamma$-category associated to a permutative category
There are several different, provably equivalent, definitions. Construction 10 in <http://www.math.uchicago.edu/~may/PAPERS/23.pdf> is one example. It is used to prove the uniqueness of a machine taking permutative categories to spectra. Therefore, any two reasonable constructions of $\Gamma$-categories from permutative categories give rise to equivalent spectra. Section 3 of Segal's original paper ``Categories and cohomology theories'' gives examples, but it is unclear
whether you are asking for examples of permutative categories (plentiful since any small symmetric monoidal category is equivalent to a permutative category) 
or of examples of different constructions of the associated $\Gamma$-category. One different, more categorical construction, is given in Theorem 3.4 of 
<http://www.math.uchicago.edu/~may/PAPERS/32.pdf>. (I should apologize for referring to my old papers since there are many other sources; that is just quickest for me.)


