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To use Python for binary linear programming to exclusively choose one group, one can follow these steps:

1. Define the decision variables as binary variables that represent whether or not a certain group is chosen.

2. Define the objective function that maximizes the value of the chosen group, while minimizing the value of the other groups.

3. Define the constraints that ensure that only one group is chosen, and that the total value of the chosen groups does not exceed a certain limit.

4. Use a linear programming solver such as PuLP or Gurobi to solve the problem and obtain the optimal solution.

Here is an example code to implement this in Python using PuLP:

from pulp import *

# Define the decision variables as binary variables
group1 = LpVariable("Group 1", cat='Binary')
group2 = LpVariable("Group 2", cat='Binary')
group3 = LpVariable("Group 3", cat='Binary')

# Define the objective function
prob = LpProblem("Exclusive Group", LpMaximize) #Maximize the chosen group, minimize the other groups
prob += 5*group1 + 2*group2 + 3*group3 # assume group1 has the highest value

# Define the constraints
prob += group1 + group2 + group3 == 1 # Only one group can be chosen
prob += 5*group1 + 2*group2 + 3*group3 <= 10 # The total value should not exceed a limit

# Solve the problem
prob.solve()

# Print the solutions
print("Group 1: ", value(group1))
print("Group 2: ", value(group2))
print("Group 3: ", value(group3))

In this example, we assume that Group 1 has the highest value, and the total value of the chosen groups should not exceed 10. The solution will show which group is chosen exclusively.