  In the paper, we introduce the terminology equivariant pointwise clutching
map. By using this, we give details on how to glue an equivariant vector bundle
over a finite set so as to obtain a new Lie group representation such that the
quotient map from the bundle to the representation is equivariant. Then, we
investigate the topology of the set of all equivariant pointwise clutching maps
with respect to an equivariant vector bundle over a finite set. Results of the
paper play a key role in classifying equivariant vector bundles over
two-surfaces in other papers.
