Mushroom production is currently a growing business in connection with wise utilization of cellulose and hemi-cellulose agricultural and food industry wastes. 

I have no idea of the type and classification of the wild edible mushrooms in Kenya, but most wild mushrooms are Agaricus and they need woody (like sawdust) kind of substrates with plenty of moisture. So for controlled cultivation, we need a dark room free of infectious microbes and other pests. Then spawn can be obtained from may be commercial suppliers or the mushroom tissues from their growth season, can be cultured and then, this can be spawned on to adequately hydrated and sterilized substrate. Then the mushrooms are allowed to grow under dark conditions for a couple of weeks (3-6) and when the mycellial mass fully covers the substrate, it has to be exposed to light to trigger spore formation (change in reproductive system) from conjugation to spore). This time the mushroom stems and heads (buttons) start forming and when they fully grow they can be harvested and processed for consumption. For more information:

http://www.un-csam.org/publication/tm-mushroom.pdf

http://library.uniteddiversity.coop/Permaculture/Mushroom_Cultivator-A_Practical_Guide_to_Growing_Mushrooms_at_Home.pdf

http://www.journeytoforever.org/farm_library/AD40.pdf

Anselimo

The first thing you need is the taxonomic identification of the mushrooms. Once you now it, you can establish (with bibliography) if they are saprophytic or ectomycorrhizal. Ectomycorrhizal ones cant be cultivated because they need their host plants to complete their life cycle. Saprotrophic ones (Agaricus, Pleurotus, Lentinula, Schizophyllum, Auricularia, etc) can be cultivated. There are other saprotrophic ones that in theory could be cultivated since they decompose organic matter, but for many of them there are not cultivation techniques developed.

Pleurotus is the fungi you can easily have in culture, Agaricus is more delicate. Try to look for native strains of Pleurotus. As I know from Niger, after a taxonomic study, some native species can be used in the genera Pleurotus, Volvariella and Agaricus.

In this link you have a chapter about mushroom cultivation in Tanzania:

http://www.karljohanstiftelsen.se/operamycologica.pdf

Best wishes