Stratified Sampling Variable Choice I am trying to do stratified sampling in R to sample from my data and one of the parameters is group, which takes variable names to sample from keeping same initial distribution of the data set.

Is it common to include the response as a grouping variable or not and what are the implications of including/excluding it?
Yes, it is very common and sometimes necessary to use the target variable for stratified sampling. 

Consider the case of fraud detection, given a bunch of features about a person (e.g. income, gender, position etc) we want to predict the likelihood of that person has committed the crime (a boolean value indicating whether the person is a suspect). This dataset is likely to be very asymmetric with very few positive examples. 

Now if we want to use k-fold cross-validation, we must stratify the samples using the target variable. If we don't, we might end up with a fold without any positive example at all and no metrics can be calculated from that fold.