Could you, please, discriminate between the shades of meaning of the two sentences below:
1. What do you like to do in your free time?
2. What do you like doing in your free time?
To my mind there is no difference between these sentences. Maybe a slight one. See below.
In general, like + ing and like + to are interchangeable when you talk about repeated actions.
If you want to be more specific, you would prefer the first sentence (infinive), such as
What do you like to do in your free time
with your friends? (expressing a particular situation)
By using the gerund, you express a generality.
Either gerund and infinitive can express a routine. The infinitive is preferable for specific situations.
However, when it comes to a situation that already exists or existed, we use -ing.
Do you like being a secretary? (you are a secretary)
The same goes for love and hate.