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In this paper, we study the complete ⋆-metric semigroups and groups and the Raǐkov completion of invariant ⋆-metric groups. We obtain the following. (1) Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>X</mi><mo>,</mo><msup><mi>d</mi><mo>⋆</mo></msup><mo>)</mo></mrow></semantics></math></inline-formula> be a complete ⋆-metric space containing a semigroup (group) <i>G</i> that is a dense subset of <i>X</i>. If the restriction of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>d</mi><mo>⋆</mo></msup></semantics></math></inline-formula> on <i>G</i> is invariant, then <i>X</i> can become a semigroup (group) containing <i>G</i> as a subgroup, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>d</mi><mo>⋆</mo></msup></semantics></math></inline-formula> is invariant on <i>X</i>. (2) Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>G</mi><mo>,</mo><msup><mi>d</mi><mo>⋆</mo></msup><mo>)</mo></mrow></semantics></math></inline-formula> be a ⋆-metric group such that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>d</mi><mo>⋆</mo></msup></semantics></math></inline-formula> is invariant on <i>G</i>. Then, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>G</mi><mo>,</mo><msup><mi>d</mi><mo>⋆</mo></msup><mo>)</mo></mrow></semantics></math></inline-formula> is complete if and only if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>G</mi><mo>,</mo><msub><mi>τ</mi><msup><mi>d</mi><mo>⋆</mo></msup></msub><mo>)</mo></mrow></semantics></math></inline-formula> is Raǐkov complete.