An investigation into one of the sub-models suggested in the type-I heavy-tailed family of distributions was carried out in this article. The model is named type-I heavy-tailed Exponential (TI-HTE) distribution. The characterizations of the model were derived and discussed. The distribution parameters were estimated utilizing the maximum likelihood approach. We further designed group acceptance sampling plans (GASP) for the rejection or otherwise of units of items in a manufacturing system based on the TI-HTE model. Simulation studies for the GASP and real-life applications were deployed highlighting the significance of the proposed model in capturing extreme behaviors beyond conventional exponential or heavy-tailed distributions.