Lightning Component:

<aura:component>
    <!--Component Start--> 
    <div class="slds-m-around_xx-large">
        <lightning:buttonGroup>
            <lightning:button label="Add" variant="Brand" onclick="{!c.handleAdd}" />
            <lightning:button label="Edit" onclick="{!c.handleEdit}" />
            <lightning:button label="Delete" disabled="true" onclick="{!c.handleDelete}" />
        </lightning:buttonGroup>
    </div>
    <!--Component End-->
</aura:component>

Lightning JS Controller:

({
    handleAdd : function(component, event, helper) {
        var selectedButtonLabel = event.getSource().get("v.label");
        alert("Button label: " + selectedButtonLabel);
    },
    
    handleEdit : function(component, event, helper) {
        var selectedButtonLabel = event.getSource().get("v.label");
        alert("Button label: " + selectedButtonLabel);
    },
    
    handleDelete : function(component, event, helper) {
        var selectedButtonLabel = event.getSource().get("v.label");
        alert("Button label: " + selectedButtonLabel);
    }
})

Output: